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

2026-02-02

AI Summary

This work proposes a novel approach based on tree decompositions for solving integer programming problems with bounded variables and constraint matrices in which each row contains at most two non-zero entries (not necessarily restricted to \{-1,0,1\}). The authors introduce a new structural parameter, total Δ-modular treewidth (TDM-treewidth), to characterize the graph-theoretic properties of such matrices and establish an analogue of the Grid Theorem for rooted signed graphs. By integrating tree decomposition, signed graph modeling, and parameterized algorithms, they design an efficient algorithm for this class of problems. This study significantly broadens the applicability of graph-structural parameters in integer programming and presents the first efficient algorithm for instances with both bounded TDM-treewidth and bounded variables.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 when variables are bounded. 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.

Problem

• integer programming
• tree decomposition
• totally Δ-modular
• graphic matrices
• bounded treewidth

Innovation

• TDM-treewidth
• integer programming
• tree decomposition
• graphic matrices
• signed graphs