Document Open Access Logo

Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM

Authors Tim Randolph , Karol Węgrzycki

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2024.96.pdf
  • Filesize: 0.88 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Parameterized algorithms
  • parameterized complexity
  • additive combinatorics
  • Subset Sum
  • integer programming
  • doubling constant

Metrics

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

Abstract

We study the parameterized complexity of algorithmic problems whose input is an integer set A in terms of the doubling constant 𝒞 := |A+A| / |A|, a fundamental measure of additive structure. We present evidence that this new parameterization is algorithmically useful in the form of new results for two difficult, well-studied problems: Integer Programming and Subset Sum.
First, we show that determining the feasibility of bounded Integer Programs is a tractable problem when parameterized in the doubling constant. Specifically, we prove that the feasibility of an integer program ℐ with n polynomially-bounded variables and m constraints can be determined in time n^{O_𝒞(1)} ⋅ poly(|ℐ|) when the column set of the constraint matrix has doubling constant 𝒞.
Second, we show that the Subset Sum and Unbounded Subset Sum problems can be solved in time n^{O_C(1)} and n^{O_𝒞(log log log n)}, respectively, where the O_C notation hides functions that depend only on the doubling constant 𝒞. We also show the equivalence of achieving an FPT algorithm for Subset Sum with bounded doubling and achieving a milestone result for the parameterized complexity of Box ILP. Finally, we design near-linear time algorithms for k-SUM as well as tight lower bounds for 4-SUM and nearly tight lower bounds for k-SUM, under the k-SUM conjecture.
Several of our results rely on a new proof that Freiman’s Theorem, a central result in additive combinatorics, can be made efficiently constructive. This result may be of independent interest.

Cite As Get BibTex

Tim Randolph and Karol Węgrzycki. Parameterized Algorithms on Integer Sets with Small Doubling: Integer Programming, Subset Sum and k-SUM. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 96:1-96:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.96

Author Details

Tim Randolph
  • Harvey Mudd College, Claremont, CA, USA
Karol Węgrzycki
  • Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

This work is a part of project TIPEA (Karol Węgrzycki) that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. and 850979).

Acknowledgements

The authors thank Lars Rohwedder for insightful discussions that helped to clarify the connections between 𝒞-Subset Sum and Hyperplane-Constrained BILP, as well as several anonymous reviewers for helpful feedback on earlier drafts.

References

  1. Amir Abboud, Karl Bringmann, and Nick Fischer. Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 391-404. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585240.
  2. Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-based Lower Bounds for Subset Sum and Bicriteria Path. ACM Transactions on Algorithms (TALG), 18(1):1-22, 2022. Google Scholar
  3. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM (JACM), 42(4):844-856, 1995. Google Scholar
  4. Per Austrin, Mikko Koivisto, Petteri Kaski, and Jesper Nederlof. Dense Subset Sum may be the hardest. 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), pages 13:1-13:14, 2016. Google Scholar
  5. Khodakhast Bibak. Additive Combinatorics: With a View Towards Computer Science and Cryptography - An Exposition. In Number Theory and Related Fields, pages 99-128, New York, NY, 2013. Springer New York. Google Scholar
  6. Karl Bringmann. A Near-Linear Pseudopolynomial Time Algorithm for Subset Sum. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1073-1084. SIAM, 2017. Google Scholar
  7. Karl Bringmann, Nick Fischer, and Vasileios Nakos. Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3069-3090. SIAM, 2022. Google Scholar
  8. 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 2021, pages 1777-1796. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.107.
  9. 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, pages 31-40, 2015. Google Scholar
  10. Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. Faster Algorithms for Bounded Knapsack and Bounded Subset Sum Via Fine-Grained Proximity Results. CoRR, abs/2307.12582, 2023. URL: https://doi.org/10.48550/arXiv.2307.12582.
  11. Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio. Subset Sum in Time 2^n/2 / poly(n), 2023. URL: https://doi.org/10.48550/arXiv.2301.07134.
  12. Jana Cslovjecsek, Friedrich Eisenbrand, Christoph Hunkenschröder, Lars Rohwedder, and Robert Weismantel. Block-Structured Integer and Linear Programming in Strongly Polynomial and Near Linear Time. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pages 1666-1681. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.101.
  13. Jana Cslovjecsek, Friedrich Eisenbrand, Michał Pilipczuk, Moritz Venzin, and Robert Weismantel. Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity. In 29th Annual European Symposium on Algorithms, ESA 2021, volume 204 of LIPIcs, pages 33:1-33:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ESA.2021.33.
  14. Jana Cslovjecsek, Martin Kouteckỳ, Alexandra Lassota, Michał Pilipczuk, and Adam Polak. Parameterized algorithms for block-structured integer programs with large entries. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 740-751. SIAM, 2024. Google Scholar
  15. Daniel Dadush, Arthur Léonard, Lars Rohwedder, and José Verschae. Optimizing Low Dimensional Functions over the Integers. In International Conference on Integer Programming and Combinatorial Optimization, pages 115-126. Springer, 2023. Google Scholar
  16. Friedrich Eisenbrand and Gennady Shmonin. Carathéodory bounds for integer cones. Oper. Res. Lett., 34(5):564-568, 2006. URL: https://doi.org/10.1016/J.ORL.2005.09.008.
  17. Friedrich Eisenbrand and Robert Weismantel. Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma. ACM Transactions on Algorithms (TALG), 16(1):1-14, 2019. Google Scholar
  18. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7:49-65, 1987. Google Scholar
  19. Gregory A Freiman. On the addition of finite sets. In Doklady Akademii Nauk, volume 158, pages 1038-1041. Russian Academy of Sciences, 1964. Google Scholar
  20. Danny Harnik and Moni Naor. On the Compressibility of NP Instances and Cryptographic Applications. SIAM Journal on Computing, 39(5):1667-1713, 2010. URL: https://doi.org/10.1137/060668092.
  21. Ellis Horowitz and Sartaj Sahni. Computing partitions with applications to the knapsack problem. Journal of the ACM (JACM), 21(2):277-292, 1974. Google Scholar
  22. Klaus Jansen and Lars Rohwedder. On integer programming and convolution. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  23. Ce Jin and Yinzhan Xu. Removing Additive Structure in 3SUM-Based Reductions. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 405-418, 2023. Google Scholar
  24. Dušan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial n-fold integer programming and applications. Math. Program., 184(1):1-34, 2020. URL: https://doi.org/10.1007/S10107-019-01402-2.
  25. Dušan Knop, Michał Pilipczuk, and Marcin Wrochna. Tight complexity lower bounds for integer linear programming with few constraints. ACM Transactions on Computation Theory (TOCT), 12(3):1-19, 2020. Google Scholar
  26. Konstantinos Koiliaris and Chao Xu. Faster pseudopolynomial time algorithms for subset sum. ACM Transactions on Algorithms (TALG), 15(3):1-20, 2019. Google Scholar
  27. Henrik W. Lenstra, Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  28. Shachar Lovett. Additive Combinatorics and its Applications in Theoretical Computer Science. Number 8 in Graduate Surveys. Theory of Computing Library, 2017. URL: https://doi.org/10.4086/toc.gs.2017.008.
  29. 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, pages 1670-1683. ACM, 2021. Google Scholar
  30. Christos H. Papadimitriou. On the complexity of integer programming. J. ACM, 28(4):765-768, 1981. URL: https://doi.org/10.1145/322276.322287.
  31. 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, volume 198 of LIPIcs, pages 106:1-106:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.106.
  32. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 974-988. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  33. Imre Z Ruzsa. Sumsets and structure. Combinatorial number theory and additive group theory, pages 87-210, 2009. Google Scholar
  34. Terence Tao and Van H Vu. Additive Combinatorics, volume 105. Cambridge University Press, 2006. Google Scholar
  35. Luca Trevisan. Additive Combinatorics and Theoretical Computer Science. ACM SIGACT News, 40:50-66, 2009. Google Scholar
  36. Emanuele Viola. Selected Results in Additive Combinatorics: An Exposition. Number 3 in Graduate Surveys. Theory of Computing Library, 2011. URL: https://doi.org/10.4086/toc.gs.2011.003.
  37. Gerhard J. Woeginger. Open problems around exact algorithms. Discrete Applied Mathematics, 156(3):397-405, 2008. Google Scholar
  38. Yufei Zhao. Graph theory and additive combinatorics. Notes for MIT, 18:49-58, 2022. Google Scholar
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