Improving proximity bounds using sparsity

TL;DR

This paper enhances bounds on the proximity between optimal solutions of integer programs and their relaxations by leveraging sparsity properties of the constraint matrix, improving understanding of solution closeness.

Contribution

The authors provide new proximity bounds based on matrix minors and entries, extending previous results and incorporating sparsity considerations for integer programs.

Findings

Proximity bounds are tight up to polynomial factors.

Bounds depend on the largest absolute value of matrix minors and entries.

Results extend to mixed integer programs.

Abstract

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.