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

TL;DR

This paper introduces a new tree-decomposition parameter called TDM-treewidth for matrices with two nonzero entries per row, enabling polynomial-time solutions for certain integer programs with bounded variables.

Contribution

It extends graph-based decomposition parameters to matrices with entries outside {-1,0,1} and establishes a Grid Theorem analogue for TDM-treewidth.

Findings

Polynomial-time algorithms for integer programs with bounded TDM-treewidth and variables.

Extension of previous graph-based parameters to matrices with arbitrary nonzero entries.

A new Grid Theorem analogue for matrices of bounded TDM-treewidth.

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.