Symmetries in Integer Programs

TL;DR

This paper explores the role of symmetries in integer programming, demonstrating that certain symmetric problems can be solved efficiently by leveraging group theory.

Contribution

It introduces a group-theoretic framework for analyzing symmetries in integer programs and shows linear-time solvability for problems with the alternating group symmetry.

Findings

Integer programs with A_n symmetry can be solved in linear time.

Symmetry analysis provides insights into the structure of integer solutions.

Group theory offers a powerful tool for understanding and solving symmetric integer programs.

Abstract

The notion of symmetry is defined in the context of Linear and Integer Programming. Symmetric integer programs are studied from a group theoretical viewpoint. We investigate the structure of integer solutions of integer programs and show that any integer program on n variables having an alternating group A_n as a group of symmetries can be solved in linear time in the number of variables.