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A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma

Authors Sushmita Gupta , Pallavi Jain , Sanjay Seetharaman , Meirav Zehavi

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Subject Classification

ACM Subject Classification
  • Theory of computation → Integer programming
Keywords
  • n-fold integer linear program
  • parameterized algorithms

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Abstract

We present an algorithm for a class of n-fold ILPs whose existing algorithms in literature are often either (1) based on the augmentation framework where one starts with an arbitrary solution and then iteratively moves towards an optimal solution by solving appropriate programs; or (2) require solving a linear relaxation of the program; or (3) are based on decomposition/proximity based arguments. Combinatorial n-fold ILPs is a class of n-fold ILPs introduced and studied by Knop et al. [MP2020] that captures several other problems in a variety of domains. We present a simple and direct algorithm that solves combinatorial n-fold ILPs with unbounded non-negative variables via an application of the Steinitz lemma. Depending on the structure of the input ILP, we also improve upon the existing algorithms in the literature in terms of the running time, thereby showing an improvement that mirrors the one shown by Rohwedder [ICALP2025] contemporaneously and independently.

Cite As Get BibTex

Sushmita Gupta, Pallavi Jain, Sanjay Seetharaman, and Meirav Zehavi. A Simple Algorithm for Combinatorial n-Fold ILPs Using the Steinitz Lemma. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.14

Author Details

Sushmita Gupta
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Pallavi Jain
  • IIT Jodhpur, India
Sanjay Seetharaman
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Acknowledgements

We thank the reviewers of an earlier draft as well as this for their contribution in improving the clarity and presentation of our results.

References

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