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Algorithms for Standard-Form ILP Problems via Komlós' Discrepancy Setting

Authors Dmitry Gribanov , Tagir Khayaleyev, Mikhail Cherniavskii, Maxim Klimenko, Dmitry Malyshev , Stanislav Moiseev

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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Parameterized complexity
  • FPT algorithms
  • Integer linear programming
  • Komlós' conjecture
  • Discrepancy

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Abstract

We study the standard-form ILP problem  c^⊤ x → max Ax = b, x ∈ ℤ_{≥ 0}ⁿ, where A ∈ ℤ^{k× n} has full row rank. We obtain refined FPT algorithms parameterized by k and Δ, the maximum absolute value of a k× k minor of A. Our approach combines discrepancy-based dynamic programming with matrix discrepancy bounds in Komlós' setting. Let κ_k denote the maximum discrepancy over all matrices with k columns whose columns have Euclidean norm at most 1. Up to polynomial factors in the input size, the optimization problem can be solved in time O(κ_k)^{2k} Δ², and the corresponding feasibility problem in time O(κ_k)^kΔ. Using the best currently known bound κ_k = Õ(log^{1/4}k), this yields running times O(log k)^{k/2(1+o(1))} Δ² and O(log k)^{k/4(1+o(1))} Δ, respectively. Under the Komlós conjecture, the dependence on k in both running times reduces to 2^O(k).

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Dmitry Gribanov, Tagir Khayaleyev, Mikhail Cherniavskii, Maxim Klimenko, Dmitry Malyshev, and Stanislav Moiseev. Algorithms for Standard-Form ILP Problems via Komlós' Discrepancy Setting. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.25

Author Details

Dmitry Gribanov
  • Laboratory LATNA, HSE University, Nizhny Novgorod, Russian Federation
  • T-Technologies, Moscow, Russian Federation
Tagir Khayaleyev
  • Moscow Independent Research Institute of Artificial Intelligence (MIRAI), Russian Federation
Mikhail Cherniavskii
  • T-Technologies, Moscow, Russian Federation
Maxim Klimenko
  • T-Technologies, Moscow, Russian Federation
Dmitry Malyshev
  • Moscow Independent Research Institute of Artificial Intelligence (MIRAI), Russian Federation
  • Lobachevsky State University of Nizhny Novgorod, Russian Federation
Stanislav Moiseev
  • T-Technologies, Moscow, Russian Federation

Funding

  • Gribanov, Dmitry: Dmitry Gribanov is partially supported in the framework of the Basic Research Program at HSE University (HSE-BR-2025-080).
  • Malyshev, Dmitry: Dmitry Malyshev is partially supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye), project No. FSMG-2024-0025.

Acknowledgements

The authors would like to thank Nikita Kasyanov, Alexander Rogozin and Demyan Yarmoshik for their participation in discussions and helpful comments. The initials LD from the definition of LD-preconditioner also serve as a quiet tribute to Liza D., who many times made difficult things feel worth doing [T. Khayaleyev]. Additionally, the authors would like to thank the anonymous reviewers for their insightful remarks and helpful suggestions.

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