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arXiv · 2602.01499

Totally $\Delta$-Modular Tree Decompositions of Graphic Matrices for Integer Programming

Caleb McFarland
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Abstract

We introduce the tree-decomposition-based parameter totally $\Delta$-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.

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Caleb McFarland. 2026-02-02. Totally $\Delta$-Modular Tree Decompositions of Graphic Matrices for Integer Programming. https://doi.org/10.4230/lipics.wg.2026.33

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