Document Open Access Logo

Revisiting the Random Subset Sum Problem

Authors Arthur Carvalho Walraven Da Cunha , Francesco d'Amore , Frédéric Giroire , Hicham Lesfari, Emanuele Natale , Laurent Viennot

PDF
Thumbnail PDF

File

LIPIcs.ESA.2023.37.pdf
  • Filesize: 0.66 MB
  • 11 pages

Document Identifiers

Author Details

Arthur Carvalho Walraven Da Cunha
  • Université Côte d'Azur, Inria Sophia Antipolis, CNRS, France
Francesco d'Amore
  • Aalto University, Finland
  • Université Côte d'Azur, Inria Sophia Antipolis, CNRS, France
Frédéric Giroire
  • Université Côte d'Azur, Inria Sophia Antipolis, CNRS, France
Hicham Lesfari
  • Université Côte d'Azur, Inria Sophia Antipolis, CNRS, France
Emanuele Natale
  • Université Côte d'Azur, Inria Sophia Antipolis, CNRS, France
Laurent Viennot
  • Inria Paris, IRIF, France

Funding

  • Giroire, Frédéric: Supported by the French government through the UCA JEDI (ANR-15-IDEX-01) and EUR DS4H (ANR-17-EURE-004) Investments in the Future projects, by the PEPR CARECLOUD, and by the European Project dAIEDGE.
  • Natale, Emanuele: Supported by the AID INRIA-DGA agreement n°2019650072.

Acknowledgements

The authors are thankful to Bianca C. Araújo and the paper reviewers for the feedback on the presentation of the paper. The authors are also grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.

Cite AsGet BibTex

Arthur Carvalho Walraven Da Cunha, Francesco d'Amore, Frédéric Giroire, Hicham Lesfari, Emanuele Natale, and Laurent Viennot. Revisiting the Random Subset Sum Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 37:1-37:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.37

Abstract

The average properties of the well-known Subset Sum Problem can be studied by means of its randomised version, where we are given a target value z, random variables X_1, …, X_n, and an error parameter ε > 0, and we seek a subset of the X_is whose sum approximates z up to error ε. In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size 𝒪(log(1/ε)) suffices to obtain, with high probability, approximations for all values in [-1/2, 1/2]. Recently, this result has been rediscovered outside the algorithms community, enabling meaningful progress in other fields. In this work, we present an alternative proof for this theorem, with a more direct approach and resourcing to more elementary tools.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Stochastic processes
  • Mathematics of computing → Combinatoric problems
Keywords
  • Random subset sum
  • Randomised method
  • Subset-sum
  • Combinatorics

Metrics

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

Related Versions

References

  1. Kazuoki Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, Second Series, 19(3):357-367, 1967. Google Scholar
  2. Luca Becchetti, Arthur Carvalho Walraven da Cunha, Andrea Clementi, Francesco d' Amore, Hicham Lesfari, Emanuele Natale, and Luca Trevisan. On the Multidimensional Random Subset Sum Problem. report, Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France ; Sapienza Università di Roma, Rome, Italy ; Università Bocconi, Milan, Italy ; Università di Roma Tor Vergata, Rome, Italy, 2022. Google Scholar
  3. Rene Beier and Berthold Vöcking. Random knapsack in expected polynomial time. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 232-241. Association for Computing Machinery, 2003. URL: https://doi.org/10.1145/780542.780578.
  4. Rene Beier and Berthold Vöcking. Probabilistic analysis of knapsack core algorithms. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '04, pages 468-477. Society for Industrial and Applied Mathematics, 2004. Google Scholar
  5. Richard Bellman. Dynamic programming. Science, 153(3731):34-37, 1966. Google Scholar
  6. Christian Borgs, Jennifer T. Chayes, and Boris G. Pittel. Phase transition and finite-size scaling for the integer partitioning problem. Random Struct. Algorithms, 19(3-4):247-288, 2001. URL: https://doi.org/10.1002/rsa.10004.
  7. Sander Borst, Daniel Dadush, Sophie Huiberts, and Danish Kashaev. A nearly optimal randomized algorithm for explorable heap selection. CoRR, abs/2210.05982, 2022. URL: https://doi.org/10.48550/arXiv.2210.05982.
  8. Sander Borst, Daniel Dadush, Sophie Huiberts, and Samarth Tiwari. On the integrality gap of binary integer programs with Gaussian data. Mathematical Programming, 2022. URL: https://doi.org/10.1007/s10107-022-01828-1.
  9. Stephen Boyd, Arpita Ghosh, Balaji Prabhakar, and Devavrat Shah. Randomized gossip algorithms. IEEE transactions on information theory, 52(6):2508-2530, 2006. Google Scholar
  10. Karl Bringmann and Philip Wellnitz. On near-linear-time algorithms for dense subset sum. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1777-1796. SIAM, 2021. Google Scholar
  11. Rebekka Burkholz, Nilanjana Laha, Rajarshi Mukherjee, and Alkis Gotovos. On the existence of universal lottery tickets. In International Conference on Learning Representations, 2022. URL: https://openreview.net/forum?id=SYB4WrJql1n.
  12. Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio. Average-case subset balancing problems. In Joseph (Seffi) Naor and Niv Buchbinder, editors, 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.
  13. Arthur da Cunha, Emanuele Natale, and Laurent Viennot. Proving the strong lottery ticket hypothesis for convolutional neural networks. In International Conference on Learning Representations, 2022. URL: https://openreview.net/forum?id=Vjki79-619-.
  14. Arthur Carvalho Walraven da Cunha, Francesco d'Amore, and Emanuele Natale. Convolutional neural networks contain structured strong lottery tickets. Preprint, June 2023. URL: https://hal.science/hal-04143024.
  15. Benjamin Doerr and Anatolii Kostrygin. Randomized rumor spreading revisited. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 138:1-138:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.138.
  16. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009. URL: http://www.cambridge.org/gb/knowledge/isbn/item2327542/.
  17. Andre Esser and Alexander May. Low weight discrete logarithms and subset sum in 2^0.65 n with polynomial memory. Cryptology ePrint Archive, 2019. Google Scholar
  18. Jonas Fischer and Rebekka Burkholz. Towards strong pruning for lottery tickets with non-zero biases. CoRR, abs/2110.11150, 2021. URL: https://arxiv.org/abs/2110.11150.
  19. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  20. Peter Gemmell and Anna M. Johnston. Analysis of a subset sum randomizer. IACR Cryptol. ePrint Arch., page 18, 2001. URL: http://eprint.iacr.org/2001/018.
  21. Brian Hayes. The easiest hard problem. American Scientist, 90:113-117, 2002. Google Scholar
  22. Alexander Helm and Alexander May. Subset sum quantumly in 1.17^n. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  23. Ce Jin, Nikhil Vyas, and Ryan Williams. Fast low-space algorithms for subset sum. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1757-1776. SIAM, 2021. Google Scholar
  24. Ce Jin and Hongxun Wu. A simple near-linear pseudopolynomial time randomized algorithm for subset sum. arXiv preprint arXiv:1807.11597, 2018. Google Scholar
  25. David S. Johnson, Cecilia R. Aragon, Lyle A. McGeoch, and Catherine Schevon. Optimization by simulated annealing: an experimental evaluation; part ii, graph coloring and number partitioning. Operations research, 39(3):378-406, 1991. Google Scholar
  26. Aniket Kate and Ian Goldberg. Generalizing cryptosystems based on the subset sum problem. Int. J. Inf. Sec., 10(3):189-199, 2011. URL: https://doi.org/10.1007/s10207-011-0129-2.
  27. George S. Lueker. Exponentially small bounds on the expected optimum of the partition and subset sum problems. Random Structures and Algorithms, 12:51-62, 1998. Google Scholar
  28. Stephan Mertens. A physicist’s approach to number partitioning. Theor. Comput. Sci., 265(1-2):79-108, 2001. URL: https://doi.org/10.1016/S0304-3975(01)00153-0.
  29. Ankit Pensia, Shashank Rajput, Alliot Nagle, Harit Vishwakarma, and Dimitris S. Papailiopoulos. Optimal lottery tickets via subset sum: Logarithmic over-parameterization is sufficient. In Hugo Larochelle, Marc'Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. URL: https://proceedings.neurips.cc/paper/2020/hash/1b742ae215adf18b75449c6e272fd92d-Abstract.html.
  30. Wheeler Ruml, J. Thomas Ngo, Joe Marks, and Stuart M Shieber. Easily searched encodings for number partitioning. Journal of Optimization Theory and Applications, 89(2):251-291, 1996. Google Scholar
  31. Zhi-Wei Sun. Unification of zero-sum problems, subset sums and covers of z. Electronic Research Announcements of The American Mathematical Society, 9:51-60, 2003. Google Scholar
  32. Chenghong Wang, Jieren Deng, Xianrui Meng, Yijue Wang, Ji Li, Sheng Lin, Shuo Han, Fei Miao, Sanguthevar Rajasekaran, and Caiwen Ding. A secure and efficient federated learning framework for NLP. In Marie-Francine Moens, Xuanjing Huang, Lucia Specia, and Scott Wen-tau Yih, editors, Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, EMNLP 2021, Virtual Event / Punta Cana, Dominican Republic, 7-11 November, 2021, pages 7676-7682. Association for Computational Linguistics, 2021. URL: https://doi.org/10.18653/v1/2021.emnlp-main.606.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact