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Fine-Grained Equivalence for Problems Related to Integer Linear Programming

Authors Lars Rohwedder , Karol Węgrzycki

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Integer Programming
  • Fine-Grained Complexity
  • Fixed-Parameter Tractable Algorithms

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Abstract

Integer Linear Programming with n binary variables and m many 0/1-constraints can be solved in time 2^Õ(m²) poly(n) and it is open whether the dependence on m is optimal. Several seemingly unrelated problems, which include variants of Closest String, Discrepancy Minimization, Set Cover, and Set Packing, can be modelled as Integer Linear Programming with 0/1 constraints to obtain algorithms with the same running time for a natural parameter m in each of the problems. Our main result establishes through fine-grained reductions that these problems are equivalent, meaning that a 2^O(m^{2-ε}) poly(n) algorithm with ε > 0 for one of them implies such an algorithm for all of them.
In the setting above, one can alternatively obtain an n^O(m) time algorithm for Integer Linear Programming using a straightforward dynamic programming approach, which can be more efficient if n is relatively small (e.g., subexponential in m). We show that this can be improved to {n'}^O(m) + O(nm), where n' is the number of distinct (i.e., non-symmetric) variables. This dominates both of the aforementioned running times.

Cite As Get BibTex

Lars Rohwedder and Karol Węgrzycki. Fine-Grained Equivalence for Problems Related to Integer Linear Programming. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 83:1-83:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.83

Author Details

Lars Rohwedder
  • University of Southern Denmark (SDU), Odense, Denmark
Karol Węgrzycki
  • Saarland University, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Rohwedder, Lars: Supported by Dutch Research Council (NWO) project “The Twilight Zone of Efficiency: Optimality of Quasi-Polynomial Time Algorithms” [grant number OCEN.W.21.268].
  • Węgrzycki, Karol: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).

Acknowledgements

We thank Friedrich Eisenbrand for his valuable insights and constructive discussions that enhanced this work.

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