Mixed Integer Reformulations of Integer Programs and the Affine TU-dimension of a Matrix

TL;DR

This paper introduces the affine TU-dimension, a generalization of total unimodularity, to reformulate integer programs with fewer integer variables, improving solution efficiency.

Contribution

It defines the affine TU-dimension, develops related theory and algorithms, and provides bounds on integer variables needed for certain integer hulls.

Findings

Reformulations with fewer integer variables are possible for certain integer programs.

The affine TU-dimension can be computed using new algorithms.

Bounds are established on the number of integer variables for specific integer hulls.

Abstract

We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the \emph{affine TU-dimension} of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls.