Document Open Access Logo

New Algorithms for Pigeonhole Equal Subset Sum

Authors Ce Jin , Ryan Williams , Stan Zhang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2025.86.pdf
  • Filesize: 0.73 MB
  • 12 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • pigeonhole principle
  • subset sums

Metrics

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

Abstract

We study the Pigeonhole Equal Subset Sum problem, which is a total-search variant of the Subset Sum problem introduced by Papadimitriou (1994): we are given a set of n positive integers {w₁,…,w_n} with the additional restriction that ∑_{i=1}^n w_i < 2ⁿ - 1, and want to find two different subsets A,B ⊆ [n] such that ∑_{i∈A} w_i = ∑_{i∈B} w_i.
Very recently, Jin and Wu (ICALP 2024) gave a randomized algorithm solving Pigeonhole Equal Subset Sum in O^*(2^{0.4n}) time, beating the classical meet-in-the-middle algorithm with O^*(2^{n/2}) runtime. In this paper, we refine Jin and Wu’s techniques to improve the runtime even further to O^*(2^{n/3}).

Cite As Get BibTex

Ce Jin, Ryan Williams, and Stan Zhang. New Algorithms for Pigeonhole Equal Subset Sum. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 86:1-86:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.86

Author Details

Ce Jin
  • MIT, Cambridge, MA, USA
Ryan Williams
  • MIT, Cambridge, MA, USA
Stan Zhang
  • MIT, Cambridge, MA, USA

Funding

  • Jin, Ce: Supported by the Jane Street Graduate Research Fellowship, NSF grant CCF-2330048, and a Simons Investigator Award.
  • Williams, Ryan: Work supported in part by NSF CCF-2420092.
  • Zhang, Stan: Self Funded

References

  1. Shyan Akmal, Lijie Chen, Ce Jin, Malvika Raj, and R. Ryan Williams. Improved merlin-arthur protocols for central problems in fine-grained complexity. Algorithmica, 85(8):2395-2426, 2023. URL: https://doi.org/10.1007/S00453-023-01102-6.
  2. Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and quantum algorithms for variants of subset-sum via dynamic programming. In 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 6:1-6:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.6.
  3. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Subset sum in the absence of concentration. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 48-61. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.48.
  4. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Dense subset sum may be the hardest. In Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS), volume 47 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.13.
  5. Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Sharper upper bounds for unbalanced uniquely decodable code pairs. IEEE Trans. Inf. Theory, 64(2):1368-1373, 2018. URL: https://doi.org/10.1109/TIT.2017.2688378.
  6. Frank Ban, Kamal Jain, Christos H. Papadimitriou, Christos-Alexandros Psomas, and Aviad Rubinstein. Reductions in PPP. Inf. Process. Lett., 145:48-52, 2019. URL: https://doi.org/10.1016/j.ipl.2018.12.009.
  7. Nikhil Bansal, Shashwat Garg, Jesper Nederlof, and Nikhil Vyas. Faster space-efficient algorithms for subset sum, k-sum, and related problems. SIAM J. Comput., 47(5):1755-1777, 2018. URL: https://doi.org/10.1137/17M1158203.
  8. Anja Becker, Jean-Sébastien Coron, and Antoine Joux. Improved generic algorithms for hard knapsacks. In Advances in Cryptology - EUROCRYPT 2011 - 30th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tallinn, Estonia, May 15-19, 2011. Proceedings, volume 6632 of Lecture Notes in Computer Science, pages 364-385. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-20465-4_21.
  9. Xavier Bonnetain, Rémi Bricout, André Schrottenloher, and Yixin Shen. Improved classical and quantum algorithms for subset-sum. In Advances in Cryptology - ASIACRYPT 2020 - 26th International Conference on the Theory and Application of Cryptology and Information Security, Daejeon, South Korea, December 7-11, 2020, Proceedings, Part II, volume 12492 of Lecture Notes in Computer Science, pages 633-666. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-64834-3_22.
  10. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. An improved pseudopolynomial time algorithm for subset sum. In 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, Chicago, IL, USA, October 27-30, 2024, pages 2202-2216. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00129.
  11. Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio. Average-case subset balancing problems. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 743-778. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.33.
  12. Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio. Subset sum in time 2^n/2 / poly(n). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11-13, 2023, Atlanta, Georgia, USA, volume 275 of LIPIcs, pages 39:1-39:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.39.
  13. Ellis Horowitz and Sartaj Sahni. Computing partitions with applications to the knapsack problem. Journal of the ACM, 21(2):277-292, 1974. URL: https://doi.org/10.1145/321812.321823.
  14. Nick Howgrave-Graham and Antoine Joux. New generic algorithms for hard knapsacks. In Advances in Cryptology - EUROCRYPT 2010, 29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Monaco / French Riviera, May 30 - June 3, 2010. Proceedings, volume 6110 of Lecture Notes in Computer Science, pages 235-256. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-13190-5_12.
  15. Ce Jin and Hongxun Wu. A faster algorithm for pigeonhole equal sums. In 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 94:1-94:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.94.
  16. Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki. Equal-subset-sum faster than the meet-in-the-middle. In 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 73:1-73:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.73.
  17. Jesper Nederlof. A short note on merlin-arthur protocols for subset sum. Inf. Process. Lett., 118:15-16, 2017. URL: https://doi.org/10.1016/j.ipl.2016.09.002.
  18. Jesper Nederlof and Karol Węgrzycki. Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectors. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1670-1683. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451024.
  19. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80063-7.
  20. Tim Randolph. Exact and Parameterized Algorithms for Subset Sum Problems. PhD thesis, Columbia University, 2024. URL: https://doi.org/10.7916/baym-5m55.
  21. Katerina Sotiraki, Manolis Zampetakis, and Giorgos Zirdelis. PPP-completeness with connections to cryptography. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 148-158. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00023.
  22. Henk C. A. van Tilborg. An upper bound for codes in a two-access binary erasure channel (corresp.). IEEE Trans. Inf. Theory, 24(1):112-116, 1978. URL: https://doi.org/10.1109/TIT.1978.1055814.
  23. Gerhard J. Woeginger. Open problems around exact algorithms. Discret. Appl. Math., 156(3):397-405, 2008. URL: https://doi.org/10.1016/J.DAM.2007.03.023.
  24. Gerhard J. Woeginger and Zhongliang Yu. On the equal-subset-sum problem. Inf. Process. Lett., 42(6):299-302, 1992. URL: https://doi.org/10.1016/0020-0190(92)90226-L.
  25. Stan Zhang. Pigeonhole Equal Subset Sum in O^*(2^n/3). Master’s thesis, Massachusetts Institute of Technology, 2024. URL: https://hdl.handle.net/1721.1/156830.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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