The Support of Integer Optimal Solutions

TL;DR

This paper establishes a bound on the support size of optimal solutions in integer linear programming problems with integral matrices, independent of the objective function, and provides a near-matching lower bound.

Contribution

The paper introduces a new bound on the support size of optimal solutions that is independent of the objective function, improving previous bounds.

Findings

Support size bound is proportional to $2m \, ext{log}(2 \, ext{sqrt}(m) \, \|A\|_\infty)$.

Provided a nearly matching asymptotic lower bound on support size.

Bound applies to any optimal solution of the integer linear program.

Abstract

The support of a vector is the number of nonzero-components. We show that given an integral $m\times n$ matrix $A$, the integer linear optimization problem $\max\left\{\boldsymbol{c}^T\boldsymbol{x} : A\boldsymbol{x} = \boldsymbol{b}, \, \boldsymbol{x}\ge\boldsymbol{0}, \,\boldsymbol{x}\in\mathbb{Z}^n\right\}$ has an optimal solution whose support is bounded by $2m \, \log (2 \sqrt{m} \| A \|_\infty)$, where $ \| A \|_\infty$ is the largest absolute value of an entry of $A$. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.