Distances of optimal solutions of mixed-integer programs

TL;DR

This paper improves bounds on the distance between optimal solutions of mixed-integer programs and their relaxations, relating it more closely to the number of integer variables and a combinatorial parameter.

Contribution

It refines existing bounds by linking the solution distance to the number of integer variables and a combinatorial parameter, using additive combinatorics techniques.

Findings

Bound on solution distance depends on $\

Feasibility of certain MILPs demonstrated using Olson's result.

Conjecture that bounds can be further improved to depend only on $\

Abstract

A classic result of Cook et al. (1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is given in terms of the number of variables and a parameter $ \Delta $, which quantifies sub-determinants of the underlying linear inequalities. We show that this distance can be bounded in terms of $ \Delta $ and the number of integer variables rather than the total number of variables. To this end, we make use of a result by Olson (1969) in additive combinatorics and demonstrate how it implies feasibility of certain mixed-integer linear programs. We conjecture that our bound can be improved to a function that only depends on $ \Delta $, in general.