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A Nearly Quadratic-Time FPTAS for Knapsack (2308.07821v3)

Published 15 Aug 2023 in cs.DS

Abstract: We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in $\widetilde{O}(n + (1/\varepsilon)2)$ time. This improves upon the $\widetilde{O}(n + (1/\varepsilon){11/5})$-time algorithm by Deng, Jin, and Mao [\textit{Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, 2023}]. Our algorithm is the best possible (up to a polylogarithmic factor) conditioned on the conjecture that $(\min, +)$-convolution has no truly subquadratic-time algorithm, since this conjecture implies that Knapsack has no $O((n + 1/\varepsilon){2-\delta})$-time FPTAS for any constant $\delta > 0$.

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References (32)
  1. Geometric applications of a matrix-searching algorithm. Algorithmica, 2(1):195–208, 1987. doi:10.1007/BF01840359.
  2. Capacitated Dynamic Programming: Faster Knapsack and Graph Algorithms. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, pages 19:1–19:13, Dagstuhl, Germany, 2019. doi:10.4230/LIPIcs.ICALP.2019.19.
  3. Faster Knapsack Algorithms via Bounded Monotone Min-Plus-Convolution. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229, pages 31:1–31:21, Dagstuhl, Germany, 2022. doi:10.4230/LIPIcs.ICALP.2022.31.
  4. Karl Bringmann. Knapsack with Small Items in Near-Quadratic Time, September 2023. arXiv:2308.03075.
  5. On Near-Linear-Time Algorithms for Dense Subset Sum. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 1777–1796. January 2021. doi:10.1137/1.9781611976465.107.
  6. Faster min-plus product for monotone instances. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022), pages 1529–1542. ACM, 2022. doi:10.1145/3519935.3520057.
  7. Solving dense subset-sum problems by using analytical number theory. Journal of Complexity, 5(3):271–282, September 1989. doi:10.1016/0885-064X(89)90025-3.
  8. Subset sums, completeness and colorings, April 2021. arXiv:2104.14766.
  9. Mark Chaimovich. New algorithm for dense subset-sum problem. In Structure theory of set addition, number 258 in Astérisque. Société mathématique de France, 1999. URL: http://www.numdam.org/item/AST_1999__258__363_0/.
  10. Timothy M. Chan. Approximation schemes for 0-1 knapsack. In Proceedings of the 1st Symposium on Simplicity in Algorithms (SOSA 2018), volume 61, pages 5:1–5:12, 2018. doi:10.4230/OASIcs.SOSA.2018.5.
  11. Faster Algorithms for Bounded Knapsack and Bounded Subset Sum Via Fine-Grained Proximity Results, July 2023. arXiv:2307.12582.
  12. On problems equivalent to (min,+)-convolution. ACM Transactions on Algorithms, 15(1):1–25, January 2019. arXiv:1702.07669, doi:10.1145/3293465.
  13. Approximating Knapsack and Partition via Dense Subset Sums. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), Proceedings, pages 2961–2979. January 2023. doi:10.1137/1.9781611977554.ch113.
  14. Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma. ACM Transactions on Algorithms, 16(1):5:1–5:14, November 2019. doi:10.1145/3340322.
  15. Gregory A Freiman. New analytical results in subset-sum problem. Discrete Mathematics, 114(1):205–217, April 1993. doi:10.1016/0012-365X(93)90367-3.
  16. An Almost Linear-Time Algorithm for the Dense Subset-Sum Problem. SIAM Journal on Computing, 20(6):1157–1189, December 1991. doi:10.1137/0220072.
  17. Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the ACM, 22(4):463–468, October 1975. doi:10.1145/321906.321909.
  18. Ce Jin. An improved FPTAS for 0-1 knapsack. In Proceedings of 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132, pages 76:1–76:14, 2019. doi:10.4230/LIPIcs.ICALP.2019.76.
  19. Ce Jin. 0-1 Knapsack in Nearly Quadratic Time, August 2023. arXiv:2308.04093.
  20. Klaus Jansen and Stefan E. J. Kraft. A faster FPTAS for the Unbounded Knapsack Problem. European Journal of Combinatorics, 68:148–174, February 2018. doi:10.1016/j.ejc.2017.07.016.
  21. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85–103. Springer US, Boston, MA, 1972. doi:10.1007/978-1-4684-2001-2_9.
  22. A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem. Journal of Combinatorial Optimization, 3(1):59–71, July 1999. doi:10.1023/A:1009813105532.
  23. Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem. Journal of Combinatorial Optimization, 8(1):5–11, March 2004. doi:10.1023/B:JOCO.0000021934.29833.6b.
  24. Knapsack Problems. Springer, 2004. doi:10.1007/978-3-540-24777-7.
  25. On the Fine-Grained Complexity of One-Dimensional Dynamic Programming. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80, pages 21:1–21:15, 2017. doi:10.4230/LIPIcs.ICALP.2017.21.
  26. Eugene L. Lawler. Fast Approximation Algorithms for Knapsack Problems. Mathematics of Operations Research, 4(4):339–356, 1979. URL: https://www.jstor.org/stable/3689221.
  27. Xiao Mao. (1−ϵ)1italic-ϵ(1-\epsilon)( 1 - italic_ϵ )-Approximation of Knapsack in Nearly Quadratic Time, August 2023. arXiv:2308.07004.
  28. Knapsack and Subset Sum with Small Items. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198, pages 106:1–106:19, Dagstuhl, Germany, 2021. doi:10.4230/LIPIcs.ICALP.2021.106.
  29. Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. PhD thesis, Massachusetts Institute of Technology, 2015. URL: https://api.semanticscholar.org/CorpusID:61901463.
  30. Sartaj Sahni. Approximate Algorithms for the 0/1 Knapsack Problem. Journal of the ACM, 22(1):115–124, January 1975. doi:10.1145/321864.321873.
  31. A. Sárközy. Finite addition theorems, I. Journal of Number Theory, 32(1):114–130, May 1989. doi:10.1016/0022-314X(89)90102-9.
  32. E. Szemerédi and V. H. Vu. Finite and Infinite Arithmetic Progressions in Sumsets. Annals of Mathematics, 163(1):1–35, 2006. URL: https://www.jstor.org/stable/20159950.
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