Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
143 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Approximating Partition in Near-Linear Time (2402.11426v2)

Published 18 Feb 2024 in cs.DS

Abstract: We propose an $\widetilde{O}(n + 1/\eps)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, the best known FPTAS for Partition runs in $\widetilde{O}(n + 1/\eps{5/4})$ time [Deng, Jin and Mao'23, Wu and Chen'22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (47)
  1. SETH-based Lower Bounds for Subset Sum and Bicriteria Path. ACM Transactions on Algorithms, 18(1):1–22, January 2022. doi:10.1145/3450524.
  2. N. Alon. Subset sums. Journal of Number Theory, 27(2):196–205, October 1987. doi:10.1016/0022-314X(87)90061-8.
  3. Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication. In Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC 2015), pages 29–36, New York, USA, June 2015. doi:10.1145/2755996.2756653.
  4. 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.
  5. Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution. In 31st Annual European Symposium on Algorithms (ESA 2023), volume 274, pages 24:1–24:16, Dagstuhl, Germany, 2023. doi:10.4230/LIPIcs.ESA.2023.24.
  6. Sparse nonnegative convolution is equivalent to dense nonnegative convolution. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021), pages 1711–1724, Virtual Italy, June 2021. doi:10.1145/3406325.3451090.
  7. Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2022), Proceedings, pages 3069–3090, January 2022. doi:10.1137/1.9781611977073.119.
  8. Top-k-convolution and the quest for near-linear output-sensitive subset sum. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), pages 982–995, 2020. doi:10.1145/3357713.3384308.
  9. Fast n-Fold Boolean Convolution via Additive Combinatorics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), 2021. doi:10.4230/LIPIcs.ICALP.2021.41.
  10. A Fine-Grained Perspective on Approximating Subset Sum and Partition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), Proceedings, pages 1797–1815, January 2021. doi:10.1137/1.9781611976465.108.
  11. Karl Bringmann. A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum. In Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 1073–1084, January 2017. doi:10.1137/1.9781611974782.69.
  12. Karl Bringmann. Knapsack with Small Items in Near-Quadratic Time, September 2023. arXiv:2308.03075.
  13. 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.
  14. Verifying candidate matches in sparse and wildcard matching. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing (STOC 2002), pages 592–601, New York, NY, USA, May 2002. doi:10.1145/509907.509992.
  15. Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In 1st Symposium on Simplicity in Algorithms (SOSA 2018), volume 61, pages 5:1–5:12, Dagstuhl, Germany, 2018. doi:10.4230/OASIcs.SOSA.2018.5.
  16. Probabilistic analysis of packing and partitioning algorithms. In Wiley-Interscience series in discrete mathematics and optimization, 1991. URL: https://api.semanticscholar.org/CorpusID:27018605.
  17. Clustered Integer 3SUM via Additive Combinatorics. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2015), pages 31–40, New York, NY, USA, June 2015. doi:10.1145/2746539.2746568.
  18. A Nearly Quadratic-Time FPTAS for Knapsack, August 2023. arXiv:2308.07821.
  19. Faster Algorithms for Bounded Knapsack and Bounded Subset Sum Via Fine-Grained Proximity Results. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2024), Proceedings, pages 4828–4848. Society for Industrial and Applied Mathematics, January 2024. doi:10.1137/1.9781611977912.171.
  20. On problems equivalent to (min,+)-convolution. ACM Transactions on Algorithms, 15(1):1–25, January 2019. doi:10.1145/3293465.
  21. 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.
  22. Approximation algorithm for some scheduling problems. Engrg. Cybernetics, 6:38–46, 1978.
  23. Computational complexity of approximation algorithms for combinatorial problems. In International Symposium on Mathematical Foundations of Computer Science, 1979. URL: https://api.semanticscholar.org/CorpusID:33715239.
  24. Fast approximation algorithms for knapsack type problems. In Optimization Techniques, Lecture Notes in Control and Information Sciences, pages 185–194, Berlin, Heidelberg, 1980. doi:10.1007/BFb0006603.
  25. A Fast Approximation Algorithm For The Subset-Sum Problem. INFOR: Information Systems and Operational Research, 32(3):143–148, 1994. doi:10.1080/03155986.1994.11732245.
  26. 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.
  27. Brian Hayes. Computing Science:The Easiest Hard Problem. American Scientist, 90(2):113–117, 2002. URL: https://www.jstor.org/stable/27857621.
  28. Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the ACM, 22(4):463–468, 1975. doi:10.1145/321906.321909.
  29. 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.
  30. Ce Jin. 0-1 Knapsack in Nearly Quadratic Time, August 2023. arXiv:2308.04093.
  31. Richard M Karp. The fast approximate solution of hard combinatorial problems. In Proc. 6th South-Eastern Conf. Combinatorics, Graph Theory and Computing (Florida Atlantic U. 1975), pages 15–31, 1975.
  32. An efficient fully polynomial approximation scheme for the Subset-Sum Problem. Journal of Computer and System Sciences, 66(2):349–370, 2003. doi:10.1016/S0022-0000(03)00006-0.
  33. 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.
  34. An efficient approximation scheme for the subset-sum problem. In Hon Wai Leong, Hiroshi Imai, and Sanjay Jain, editors, Algorithms and Computation, Lecture Notes in Computer Science, pages 394–403, Berlin, Heidelberg, 1997. doi:10.1007/3-540-63890-3_42.
  35. 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.
  36. 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.
  37. Xiao Mao. (1−ϵ)1italic-ϵ(1-\epsilon)( 1 - italic_ϵ )-Approximation of Knapsack in Nearly Quadratic Time, August 2023. arXiv:2308.07004.
  38. Hiding information and signatures in trapdoor knapsacks. IEEE Transactions on Information Theory, 24(5):525–530, September 1978. doi:10.1109/TIT.1978.1055927.
  39. A Subquadratic Approximation Scheme for Partition. In Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), Proceedings, pages 70–88, January 2019. doi:10.1137/1.9781611975482.5.
  40. Vasileios Nakos. Nearly Optimal Sparse Polynomial Multiplication. IEEE Transactions on Information Theory, 66(11):7231–7236, November 2020. doi:10.1109/TIT.2020.2989385.
  41. 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.
  42. Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. PhD thesis, Massachusetts Institute of Technology, 2015. URL: https://api.semanticscholar.org/CorpusID:61901463.
  43. A. Sárközy. Finite addition theorems, I. Journal of Number Theory, 32(1):114–130, May 1989. doi:10.1016/0022-314X(89)90102-9.
  44. A. Sárközy. Fine Addition Theorems, II. Journal of Number Theory, 48(2):197–218, August 1994. doi:10.1006/jnth.1994.1062.
  45. E. Szemerédi and V. Vu. Long arithmetic progressions in sumsets: Thresholds and bounds. Journal of the American Mathematical Society, 19(1):119–169, September 2005. doi:10.1090/S0894-0347-05-00502-3.
  46. 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.
  47. Improved Approximation Schemes for (Un-)Bounded Subset-Sum and Partition, December 2022. arXiv:2212.02883.
Citations (6)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets