On the size of integer programs with bounded non-vanishing subdeterminants

TL;DR

This paper investigates the maximum number of columns in integer matrices with fixed row rank and bounded non-zero minors, providing new bounds and constructions relevant to integer programming complexity.

Contribution

It introduces new linear and asymptotic bounds on the column count for matrices with bounded subdeterminants, using coding theory and geometry of numbers techniques.

Findings

Linear upper bound for r=2 matches lower bound constructions.

Asymptotic sublinear bounds derived for general r.

Discussion of computational methods for small parameters.

Abstract

Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the column number question for integer matrices whose every $r \times r$ minor is non-zero and bounded by a fixed constant $\Delta$ in absolute value. Approaching the problem in two different ways, one that uses results from coding theory, and the other from the geometry of numbers, we obtain linear and asymptotically sublinear upper bounds on the maximal number of columns of such matrices, respectively. We complement these results by lower bound constructions, matching the linear upper bound for $r=2$, and a discussion of a computational approach to determine the maximal number of columns for small parameters $\Delta$ and $r$.