Computing symmetry groups of polyhedra

TL;DR

This paper presents algorithms for computing various symmetry groups of polyhedra, including linear, projective, and combinatorial symmetries, with practical applications in graph automorphisms and integer programming.

Contribution

It introduces new algorithmic methods for computing symmetry groups of polyhedra and discusses their practical implementation and applications.

Findings

Linear symmetry group computation reduces to graph automorphism problems

Algorithms for projective and combinatorial symmetry groups are developed

Practical experiences demonstrate effectiveness of the methods

Abstract

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used for instance in integer linear programming.