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

Caleb McFarland

TL;DR

This work introduces the totally Δ-modular treewidth (TDM-treewidth) as a parameter for matrices with two nonzero entries per row, unifying signed-graph and rooted-graph perspectives. It establishes a polynomial-time solvability framework for integer programs when the matrix has bounded TDM-treewidth and the decision variables lie in a bounded interval, with a runtime of the form d^{O(k)} n^{f(k)}. Extending beyond coefficients in {−1,0,1}, the authors connect TDM-treewidth to K-free treewidth and tame OCP-treewidth, providing a grid-minor structure theorem via rooted signed graphs and a corresponding DP-based algorithm. The paper also presents an approximate grid theorem and shows how these decompositions yield tractable IPs for a broad class of totally Δ-modular matrices, highlighting deep connections to MIS in odd-minor-closed classes and outlining several future avenues, including relaxations for {0,1}-variables and bounded-entry versus unbounded-variable regimes.

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 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.

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

TL;DR

This work introduces the totally Δ-modular treewidth (TDM-treewidth) as a parameter for matrices with two nonzero entries per row, unifying signed-graph and rooted-graph perspectives. It establishes a polynomial-time solvability framework for integer programs when the matrix has bounded TDM-treewidth and the decision variables lie in a bounded interval, with a runtime of the form d^{O(k)} n^{f(k)}. Extending beyond coefficients in {−1,0,1}, the authors connect TDM-treewidth to K-free treewidth and tame OCP-treewidth, providing a grid-minor structure theorem via rooted signed graphs and a corresponding DP-based algorithm. The paper also presents an approximate grid theorem and shows how these decompositions yield tractable IPs for a broad class of totally Δ-modular matrices, highlighting deep connections to MIS in odd-minor-closed classes and outlining several future avenues, including relaxations for {0,1}-variables and bounded-entry versus unbounded-variable regimes.

Abstract

We introduce the tree-decomposition-based parameter totally -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 . 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.
Paper Structure (13 sections, 23 theorems, 15 equations, 2 figures)

This paper contains 13 sections, 23 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signed graphs
  4. Odd-minors and signed graphs
  5. Rooted (signed) graphs
  6. Matrices
  7. K-free treewidth
  8. (tame) OCP-treewidth
  9. Properties of TDM-treewidth
  10. Solving integer programs with large entries
  11. Further discussion
  12. On {0,1}-variables and the definition of TDM-treewidth
  13. Bounded entries and unbounded variables

Key Result

Theorem 1.2

Let $d,k$ be nonnegative integers, and let $A$ be a matrix with two nonzero entries per row whose associated rooted signed graph has $\textsf{TDM}$-treewidth at most $k$. Then for any $w \in \mathbb{Z}^n, b \in \mathbb{Z}^m$, and $\ell,u \in \mathbb{Z}^m$ with $\|u - \ell\|_\infty \leq d$, we can so in $d^{\mathcal{O}(k)}n^{f(k)}$ time for some computable $f$.

Figures (2)

  • Figure 1: The construction in the proof of \ref{['lem:totallyevengrid']}.
  • Figure :

Theorems & Definitions (47)

  • Conjecture 1.1: shevchenko1996qualitative
  • Definition 1.2: TDM-treewidth
  • Theorem 1.2
  • Definition 1.3: Parity Handle $\mathcal{H}_k$ and parity vortex $\mathcal{V}_k$
  • Definition 1.4: Rooted Grid $\mathcal{W}_k$
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1: tree decomposition
  • Definition 2.2: grid
  • Lemma 2.3
  • ...and 37 more