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

Authors Tim Randolph , Karol Węgrzycki

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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.

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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

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.

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

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