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Practical and Parallelizable Algorithms for Non-Monotone Submodular Maximization with Size Constraint (2009.01947v5)

Published 3 Sep 2020 in cs.DS and cs.LG

Abstract: We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to $0.193 - \varepsilon$. The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in $O( \log(n) )$ adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio $1/6 - \varepsilon$, adaptivity $O( \log (n) )$, and query complexity $O( n \log (k) )$, while the second has approximation ratio $0.193 - \varepsilon$, adaptivity $O( \log2 (n) )$, and query complexity $O(n \log (k))$. Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

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References (34)
  1. Submodular maximization subject to a knapsack constraint: Combinatorial algorithms with near-optimal adaptive complexity.  In Meila, M.,  and Zhang, T. (Eds.), Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, Vol. 139 of Proceedings of Machine Learning Research, pp. 231–242. PMLR.
  2. Fast algorithms for maximizing submodular functions.  In Chekuri, C. (Ed.), Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pp. 1497–1514. SIAM.
  3. Non-monotone submodular maximization in exponentially fewer iterations.  In Bengio, S., Wallach, H. M., Larochelle, H., Grauman, K., Cesa-Bianchi, N., and Garnett, R. (Eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pp. 2359–2370.
  4. An exponential speedup in parallel running time for submodular maximization without loss in approximation.  In Chan, T. M. (Ed.), Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pp. 283–302. SIAM.
  5. An optimal approximation for submodular maximization under a matroid constraint in the adaptive complexity model.  In Charikar, M.,  and Cohen, E. (Eds.), Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pp. 66–77. ACM.
  6. The adaptive complexity of maximizing a submodular function.  In Diakonikolas, I., Kempe, D., and Henzinger, M. (Eds.), Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pp. 1138–1151. ACM.
  7. Constrained submodular maximization via a nonsymmetric technique.  Math. Oper. Res., 44(3), 988–1005.
  8. A tight linear time (1/2)-approximation for unconstrained submodular maximization.  In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pp. 649–658. IEEE Computer Society.
  9. Comparing apples and oranges: Query tradeoff in submodular maximization.  In Indyk, P. (Ed.), Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pp. 1149–1168. SIAM.
  10. Parallelizing greedy for submodular set function maximization in matroids and beyond.  In Charikar, M.,  and Cohen, E. (Eds.), Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pp. 78–89. ACM.
  11. Unconstrained submodular maximization with constant adaptive complexity.  In Charikar, M.,  and Cohen, E. (Eds.), Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pp. 102–113. ACM.
  12. Best of both worlds: Practical and theoretically optimal submodular maximization in parallel.  In Ranzato, M., Beygelzimer, A., Dauphin, Y. N., Liang, P., and Vaughan, J. W. (Eds.), Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pp. 25528–25539.
  13. Beyond keyword search: discovering relevant scientific literature.  In Apté, C., Ghosh, J., and Smyth, P. (Eds.), Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Diego, CA, USA, August 21-24, 2011, pp. 439–447. ACM.
  14. Submodular maximization with nearly-optimal approximation and adaptivity in nearly-linear time.  In Chan, T. M. (Ed.), Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pp. 274–282. SIAM.
  15. Parallel algorithm for non-monotone dr-submodular maximization.  In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, Vol. 119 of Proceedings of Machine Learning Research, pp. 2902–2911. PMLR.
  16. Submodular maximization with matroid and packing constraints in parallel.  In Charikar, M.,  and Cohen, E. (Eds.), Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pp. 90–101. ACM.
  17. Submodular Maximization with Nearly Optimal Approximation, Adaptivity, and Query Complexity.  In ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 255–273.
  18. Non-monotone submodular maximization with nearly optimal adaptivity and query complexity.  arXiv preprint arXiv:1808.06932.
  19. Non-monotone submodular maximization with nearly optimal adaptivity and query complexity.  In Chaudhuri, K.,  and Salakhutdinov, R. (Eds.), Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, Vol. 97 of Proceedings of Machine Learning Research, pp. 1833–1842. PMLR.
  20. Maximizing non-monotone submodular functions.  SIAM J. Comput., 40(4), 1133–1153.
  21. Submodular maximization by simulated annealing.  In Randall, D. (Ed.), Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pp. 1098–1116. SIAM.
  22. Constrained non-monotone submodular maximization: Offline and secretary algorithms.  In Saberi, A. (Ed.), Internet and Network Economics - 6th International Workshop, WINE 2010, Stanford, CA, USA, December 13-17, 2010. Proceedings, Vol. 6484 of Lecture Notes in Computer Science, pp. 246–257. Springer.
  23. Optimal marketing strategies over social networks.  In Huai, J., Chen, R., Hon, H., Liu, Y., Ma, W., Tomkins, A., and Zhang, X. (Eds.), Proceedings of the 17th International Conference on World Wide Web, WWW 2008, Beijing, China, April 21-25, 2008, pp. 189–198. ACM.
  24. Submodular streaming in all its glory: Tight approximation, minimum memory and low adaptive complexity.  In Chaudhuri, K.,  and Salakhutdinov, R. (Eds.), Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, Vol. 97 of Proceedings of Machine Learning Research, pp. 3311–3320. PMLR.
  25. Maximizing the spread of influence through a social network.  In Getoor, L., Senator, T. E., Domingos, P. M., and Faloutsos, C. (Eds.), Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Washington, DC, USA, August 24 - 27, 2003, pp. 137–146. ACM.
  26. Kuhnle, A. (2021). Nearly linear-time, parallelizable algorithms for non-monotone submodular maximization.  In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Virtual Event, February 2-9, 2021, pp. 8200–8208. AAAI Press.
  27. Choosing non-redundant representative subsets of protein sequence data sets using submodular optimization.  In Shehu, A., Wu, C. H., Boucher, C., Li, J., Liu, H., and Pop, M. (Eds.), Proceedings of the 2018 ACM International Conference on Bioinformatics, Computational Biology, and Health Informatics, BCB 2018, Washington, DC, USA, August 29 - September 01, 2018, p. 566. ACM.
  28. Fast constrained submodular maximization: Personalized data summarization.  In Balcan, M.,  and Weinberger, K. Q. (Eds.), Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, Vol. 48 of JMLR Workshop and Conference Proceedings, pp. 1358–1367. JMLR.org.
  29. Growth of the flickr social network.  In Faloutsos, C., Karagiannis, T., and Rodriguez, P. (Eds.), Proceedings of the first Workshop on Online Social Networks, WOSN 2008, Seattle, WA, USA, August 17-22, 2008, pp. 25–30. ACM.
  30. Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis. Cambridge university press.
  31. Best algorithms for approximating the maximum of a submodular set function.  Math. Oper. Res., 3(3), 177–188.
  32. Scene summarization for online image collections.  In IEEE 11th International Conference on Computer Vision, ICCV 2007, Rio de Janeiro, Brazil, October 14-20, 2007, pp. 1–8. IEEE Computer Society.
  33. Temporal corpus summarization using submodular word coverage.  In Chen, X., Lebanon, G., Wang, H., and Zaki, M. J. (Eds.), 21st ACM International Conference on Information and Knowledge Management, CIKM’12, Maui, HI, USA, October 29 - November 02, 2012, pp. 754–763. ACM.
  34. Learning mixtures of submodular functions for image collection summarization.  In Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N. D., and Weinberger, K. Q. (Eds.), Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pp. 1413–1421.
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