Outer approximations of core points for integer programming

TL;DR

This paper introduces new outer approximation methods for symmetric integer linear programs that improve efficiency by leveraging the structure of symmetry groups, especially those with large disjoint cycles.

Contribution

It develops novel techniques for solving symmetric ILPs using outer approximations of core points, removing the need for finiteness of core points and enhancing efficiency with certain symmetry groups.

Findings

Methods are more efficient for groups with large disjoint cycles.

Outer approximations can solve symmetric ILPs without finiteness assumptions.

Approach generalizes core point techniques beyond finite cases.

Abstract

For several decades the dominant techniques for integer linear programming have been branching and cutting planes. Recently, several authors have developed core point methods for solving symmetric integer linear programs (ILPs). An integer point is called a core point if its orbit polytope is lattice-free. It has been shown that for symmetric ILPs, optimizing over the set of core points gives the same answer as considering the entire space. Existing core point techniques rely on the number of core points (or equivalence classes) being finite, which requires special symmetry groups. In this paper we develop some new methods for solving symmetric ILPs (based on outer approximations of core points) that do not depend on finiteness but are more efficient if the group has large disjoint cycles in its set of generators.