Algorithms for Highly Symmetric Linear and Integer Programs

TL;DR

This paper presents methods to leverage symmetry in linear and integer programming, reducing problem complexity and developing an efficient algorithm for highly symmetric cases that is linear in constraints and quadratic in dimension.

Contribution

It introduces a novel approach combining symmetry properties and geometric insights to efficiently solve highly symmetric integer linear programs.

Findings

Reduced problem dimension using symmetry properties

Algorithm runs in linear time relative to constraints

Quadratic time complexity in the problem dimension

Abstract

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.