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Totally Δ-Modular Tree Decompositions of Graphic Matrices for Integer Programming

Author Caleb McFarland

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LIPIcs.WG.2026.33.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Mathematics of computing → Graph algorithms
Keywords
  • Integer programming
  • subdeterminants
  • independent set
  • rooted graphs
  • signed graphs
  • odd cycle packing number

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Abstract

We introduce the tree-decomposition-based parameter totally Δ-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth in polynomial time when variables have bounded domain. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of {-1,0,1}. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.

Cite As Get BibTex

Caleb McFarland. Totally Δ-Modular Tree Decompositions of Graphic Matrices for Integer Programming. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.33

Author Details

Caleb McFarland
  • Georgia Institute of Technology, Atlanta, GA, USA

Funding

Supported by the National Science Foundation under Grant No. DMS-2452111 and by the Georgia Tech ARC-ACO Fellowship.

Acknowledgements

The author would like to thank Maximilian Gorsky, Rose McCarty, and Sebastian Wiederrecht for valuable discussions.

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