Notes on $\{a,b,c\}$-Modular Matrices

TL;DR

This paper develops a polynomial time algorithm to recognize $ ext{\{a,b,c\}}$-modular matrices and solve related integer programming problems, extending the recognition of totally unimodular matrices to a broader class.

Contribution

It introduces a polynomial time recognition algorithm for $ ext{\{a,b,c\}}$-modular matrices, generalizing totally unimodular matrix recognition and enabling efficient solutions for associated integer programs.

Findings

Recognition is polynomial time unless a duplicative relation exists.

Provides a polynomial time algorithm for integer programming with $ ext{\{a,b,c\}}$-modular matrices.

Extends the class of matrices for which integer programming can be efficiently solved.

Abstract

Let $A \in \mathbb{Z}^{m \times n}$ be an integral matrix and $a$, $b$, $c \in \mathbb{Z}$ satisfy $a \geq b \geq c \geq 0$. The question is to recognize whether $A$ is $\{a,b,c\}$-modular, i.e., whether the set of $n \times n$ subdeterminants of $A$ in absolute value is $\{a,b,c\}$. We will succeed in solving this problem in polynomial time unless $A$ possesses a duplicative relation, that is, $A$ has nonzero $n \times n$ subdeterminants $k_1$ and $k_2$ satisfying $2 \cdot |k_1| = |k_2|$. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over $\{a,b,c\}$-modular constraint matrices for any constants $a$, $b$ and $c$.