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On Matrices over a Polynomial Ring with Restricted Subdeterminants

Marcel Celaya, Stefan Kuhlmann, Robert Weismantel

TL;DR

The paper develops a framework for parametric integer optimization where constraint matrices lie in $\mathbb{Z}[x]$ and subdeterminants are restricted to a finite set $S$, introducing totally $S$-modular matrices and forbidden minors. It proves that, when $S$ is finite, the determinants of forbidden minors form a finite set, and derives polynomial-time recognition and ILP solvability results for key specialized families after evaluating polynomials at integers, notably for $S(a)=\pm\{0,1,a,a+1,2a+1\}$ and $S(a)=\pm\{0,a,a+1,2a+1\}$ under suitable exclusions. A central tool is the decomposition $\bm{M}=\bm{T}+x\bm{u}\bm{v}^\top$ (and its generalizations), which reduces modularity checks to total unimodularity tests on $\bm{T}$ and $\bm{T}-\bm{u}\bm{v}^\top$ and enables determinant-based reasoning via identities such as the Desnanot–Jacobi rule and the matrix determinant lemma. The work connects combinatorial matrix structure to tractable optimization in a parametric setting, while clarifying the limits of finiteness for forbidden minors and outlining open questions about when such lists are finite. These results offer a pathway to polynomial-time verification and optimization for a broad class of parametric integer programs with structured polynomial-parameter matrices, and motivate further study of the forbidden-minor landscape in this algebraic context.

Abstract

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We investigate in particular matrices whose subdeterminants all lie in a fixed set $S\subseteq\mathbb{Z}[x]$. Such matrices, which we call totally $S$-modular matrices, are closed with respect to taking submatrices, so it is natural to look at minimally non-totally $S$-modular matrices which we call forbidden minors for $S$. Among other results, we prove that if $S$ is finite, then the set of all determinants attained by a forbidden minor for $S$ is also finite. Specializing to the integers, we subsequently obtain the following positive complexity result: the recognition problem for totally $\pm\{0,1,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{-3,-2,1,2\}$ and the integer linear optimization problem for totally $\pm\{ 0,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{ -2,1\}$ can be solved in polynomial time.

On Matrices over a Polynomial Ring with Restricted Subdeterminants

TL;DR

The paper develops a framework for parametric integer optimization where constraint matrices lie in and subdeterminants are restricted to a finite set , introducing totally -modular matrices and forbidden minors. It proves that, when is finite, the determinants of forbidden minors form a finite set, and derives polynomial-time recognition and ILP solvability results for key specialized families after evaluating polynomials at integers, notably for and under suitable exclusions. A central tool is the decomposition (and its generalizations), which reduces modularity checks to total unimodularity tests on and and enables determinant-based reasoning via identities such as the Desnanot–Jacobi rule and the matrix determinant lemma. The work connects combinatorial matrix structure to tractable optimization in a parametric setting, while clarifying the limits of finiteness for forbidden minors and outlining open questions about when such lists are finite. These results offer a pathway to polynomial-time verification and optimization for a broad class of parametric integer programs with structured polynomial-parameter matrices, and motivate further study of the forbidden-minor landscape in this algebraic context.

Abstract

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring of polynomials in one variable. We investigate in particular matrices whose subdeterminants all lie in a fixed set . Such matrices, which we call totally -modular matrices, are closed with respect to taking submatrices, so it is natural to look at minimally non-totally -modular matrices which we call forbidden minors for . Among other results, we prove that if is finite, then the set of all determinants attained by a forbidden minor for is also finite. Specializing to the integers, we subsequently obtain the following positive complexity result: the recognition problem for totally -modular matrices with and the integer linear optimization problem for totally -modular matrices with can be solved in polynomial time.
Paper Structure (10 sections, 14 theorems, 29 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The smallest non-trivial cases
  3. Statements of our results
  4. Tools
  5. Properties of $I(S)$
  6. Proofs of Theorems \ref{['thm_recognition']} and \ref{['thm_opt_a_a_a+1']}
  7. Proof of Theorem \ref{['thm_recognition']}
  8. Proof of Theorem \ref{['thm_opt_a_a_a+1']}
  9. Finiteness of forbidden minors
  10. Acknowledgements.

Key Result

theorem thmcountertheorem

Let $S\subseteq \mathbb{Z}[x]$ be finite. Let $\bm{M}=\bm{T} + x\cdot\bm{u}\cdot\bm{v}^\top$ where $\bm{T},\bm{u},$ and $\bm{v}$ are integral. If $\bm{M}$ is totally $S$-modular, then $\bm{T}$ is totally $S(0)$-modular.

Figures (1)

  • Figure 1: The blue boxes depict the value $x$ and the green boxes $x+1$ or vice versa. The first row of matrices corresponds to the first five elements of an infinite sequence of matrices that can be obtained by generalizing the existing pattern. It can be shown that those infinitely many matrices are forbidden minors for $\pm\lbrace 0,1,x,x+1,2x+1\rbrace$. The matrices in the last row correspond to five forbidden minors for $\pm \lbrace 0,x,x+1,2x+1\rbrace$.

Theorems & Definitions (26)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • lemma thmcounterlemma: Sylvester's determinant identity
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • ...and 16 more