Document Open Access Logo

Weakly Approximating Knapsack in Subquadratic Time

Authors Lin Chen , Jiayi Lian , Yuchen Mao , Guochuan Zhang

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.51.pdf
  • Filesize: 0.75 MB
  • 18 pages
HTML5 Logo HTML (experimental)
LIPIcs.ICALP.2025.51.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Knapsack
  • FPTAS

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We consider the classic Knapsack problem. Let t and OPT be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least OPT/(1 + ε) and total weight at most t, then Knapsack can be solved in Õ(n + (1/(ε))²) time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that (min,+)-convolution cannot be solved in truly subquadratic time [Künnemann, Paturi, and Schneider '17][Cygan, Mucha, Węgrzycki, and Włodarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least OPT and total weight at most (1 + ε)t. Therefore, it is natural to ask the following question. 
If one seeks a solution with total profit at least OPT/(1+ε) and total weight at most (1 + ε)t, can Knsapck be solved in Õ(n + (1/(ε))^{2-δ}) time for some constant δ > 0? 
We answer this open question affirmatively by proposing an Õ(n + (1/(ε))^{7/4})-time algorithm.

Cite As Get BibTex

Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. Weakly Approximating Knapsack in Subquadratic Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.51

Author Details

Lin Chen
  • Zhejiang University, Hangzhou, China
Jiayi Lian
  • Zhejiang University, Hangzhou, China
Yuchen Mao
  • Zhejiang University, Hangzhou, China
Guochuan Zhang
  • Zhejiang University, Hangzhou, China

Funding

  • Mao, Yuchen: National Natural Science Foundation of China [Project No. 62402436].
  • Zhang, Guochuan: National Natural Science Foundation of China [Project No. 12271477].

References

  1. Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2(1):195-208, November 1987. URL: https://doi.org/10.1007/BF01840359.
  2. Kyriakos Axiotis and Christos Tzamos. Capacitated Dynamic Programming: Faster Knapsack and Graph Algorithms. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), pages 19:1-19:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.19.
  3. David Bremner, Timothy M Chan, Erik D Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Pǎtraşcu, and Perouz Taslakian. Necklaces, convolutions, and x+ y. Algorithmica, 69(2):294-314, 2014. URL: https://doi.org/10.1007/S00453-012-9734-3.
  4. 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. Society for Industrial and Applied Mathematics, 2017. URL: https://doi.org/10.1137/1.9781611974782.69.
  5. Karl Bringmann. Knapsack with small items in near-quadratic time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 259-270. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649719.
  6. Karl Bringmann and Alejandro Cassis. Faster Knapsack Algorithms via Bounded Monotone Min-Plus-Convolution. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), pages 31:1-31:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.31.
  7. Karl Bringmann and Alejandro Cassis. Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution. In 31st Annual European Symposium on Algorithms (ESA 2023), pages 24:1-24:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.24.
  8. Karl Bringmann, Anita Dürr, and Adam Polak. Even faster knapsack via rectangular monotone min-plus convolution and balancing. In Timothy M. Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms, ESA 2024, September 2-4, 2024, Royal Holloway, London, United Kingdom, volume 308 of LIPIcs, pages 33:1-33:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ESA.2024.33.
  9. Karl Bringmann and Vasileios Nakos. A Fine-Grained Perspective on Approximating Subset Sum and Partition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 1797-1815. Society for Industrial and Applied Mathematics, 2021. URL: https://doi.org/10.1137/1.9781611976465.108.
  10. Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In 1st Symposium on Simplicity in Algorithms (SOSA 2018), pages 5:1-5:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/OASIcs.SOSA.2018.5.
  11. Timothy M. Chan and Moshe Lewenstein. Clustered Integer 3SUM via Additive Combinatorics. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2015), pages 31-40. Association for Computing Machinery, 2015. URL: https://doi.org/10.1145/2746539.2746568.
  12. Timothy M. Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Proceedings of the 2016 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pages 1246-1255, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch87.
  13. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. Approximating partition in near-linear time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 307-318. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649727.
  14. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. 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), pages 4828-4848. Society for Industrial and Applied Mathematics, 2024. URL: https://doi.org/10.1137/1.9781611977912.171.
  15. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. A nearly quadratic-time fptas for knapsack. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 283-294. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649730.
  16. Shucheng Chi, Ran Duan, Tianle Xie, and Tianyi Zhang. 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. Association for Computing Machinery, 2022. URL: https://doi.org/10.1145/3519935.3520057.
  17. Marek Cygan, Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. On problems equivalent to (min,+)-convolution. ACM Transactions on Algorithms, 15(1):1-25, January 2019. URL: https://doi.org/10.1145/3293465.
  18. Mingyang Deng, Ce Jin, and Xiao Mao. Approximating Knapsack and Partition via Dense Subset Sums. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023), pages 2961-2979. Society for Industrial and Applied Mathematics, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch113.
  19. Friedrich Eisenbrand and Robert Weismantel. Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma. ACM Transactions on Algorithms, 16(1):5:1-5:14, November 2019. URL: https://doi.org/10.1145/3340322.
  20. Aleksei V. Fishkin, Olga Gerber, Klaus Jansen, and Roberto Solis-Oba. On packing squares with resource augmentation: maximizing the profit. In Proceedings of the 2005 Australasian Symposium on Theory of Computing, CATS '05, pages 61-67, AUS, 2005. Australian Computer Society, Inc. URL: https://dl.acm.org/doi/10.5555/1082260.1082268.
  21. Oscar H. Ibarra and Chul E. Kim. Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the ACM, 22(4):463-468, October 1975. URL: https://doi.org/10.1145/321906.321909.
  22. Kazuo Iwama and Guochuan Zhang. Online knapsack with resource augmentation. Information Processing Letters, 110(22):1016-1020, 2010. URL: https://doi.org/10.1016/j.ipl.2010.08.013.
  23. Klaus Jansen and Stefan E. J. Kraft. A faster FPTAS for the Unbounded Knapsack Problem. European Journal of Combinatorics, 68:148-174, February 2018. URL: https://doi.org/10.1016/j.ejc.2017.07.016.
  24. Ce Jin. An improved FPTAS for 0-1 knapsack. In Proceedings of 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), pages 76:1-76:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.76.
  25. Ce Jin. 0-1 Knapsack in Nearly Quadratic Time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 271-282. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649618.
  26. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Springer US, Boston, MA, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  27. Hans Kellerer, Renata Mansini, Ulrich Pferschy, and Maria Grazia Speranza. An efficient fully polynomial approximation scheme for the Subset-Sum Problem. Journal of Computer and System Sciences, 66(2):349-370, March 2003. URL: https://doi.org/10.1016/S0022-0000(03)00006-0.
  28. Hans Kellerer and Ulrich Pferschy. A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem. Journal of Combinatorial Optimization, 3(1):59-71, July 1999. URL: https://doi.org/10.1023/A:1009813105532.
  29. Hans Kellerer and Ulrich Pferschy. Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem. Journal of Combinatorial Optimization, 8(1):5-11, March 2004. URL: https://doi.org/10.1023/B:JOCO.0000021934.29833.6b.
  30. Hans Kellerer, Ulrich Pferschy, and Pisinger David. Knapsack Problems. Springer, Berlin Heidelberg, 2004. URL: https://doi.org/10.1007/978-3-540-24777-7.
  31. Konstantinos Koiliaris and Chao Xu. Faster Pseudopolynomial Time Algorithms for Subset Sum. ACM Transactions on Algorithms, 15(3):1-20, July 2019. URL: https://doi.org/10.1145/3329863.
  32. Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the Fine-Grained Complexity of One-Dimensional Dynamic Programming. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), pages 21:1-21:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.21.
  33. Eugene L. Lawler. Fast Approximation Algorithms for Knapsack Problems. Mathematics of Operations Research, 4(4):339-356, November 1979. URL: https://doi.org/10.1287/MOOR.4.4.339.
  34. Xiao Mao. (1-ε)-approximation of knapsack in nearly quadratic time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 295-306. Association for Computing Machinery, 2024. URL: https://doi.org/10.1145/3618260.3649677.
  35. Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. A Subquadratic Approximation Scheme for Partition. In Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 70-88. Society for Industrial and Applied Mathematics, 2019. URL: https://doi.org/10.1137/1.9781611975482.5.
  36. Adam Polak, Lars Rohwedder, and Karol Węgrzycki. Knapsack and Subset Sum with Small Items. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), pages 106:1-106:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.106.
  37. Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. PhD thesis, Massachusetts Institute of Technology, 2015. URL: https://dspace.mit.edu/handle/1721.1/98564.
  38. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing (STOC 2014), pages 664-673. Association for Computing Machinery, 2014. URL: https://doi.org/10.1145/2591796.2591811.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact