Breaking Symmetries from a Set-Covering Perspective

TL;DR

This paper presents a novel set-covering formulation for symmetry breaking in graphs, enabling optimal and partial symmetry breaks that improve existing methods, with potential for leveraging extensive set-covering research.

Contribution

It introduces a set-covering perspective to symmetry breaking, allowing for optimal solutions and partial breaks that outperform current approaches.

Findings

Achieved optimal LexLeader symmetry breaks for graphs up to order 10.

Developed partial symmetry breaks that surpass state-of-the-art.

Connected symmetry breaking with decades of set-covering research.

Abstract

We formalize symmetry breaking as a set-covering problem. For the case of breaking symmetries on graphs, a permutation covers a graph if applying it to the graph yields a smaller graph in a given order. Canonical graphs are those that cannot be made smaller by any permutation. A complete symmetry break is then a set of permutations that covers all non-canonical graphs. A complete symmetry break with a minimal number of permutations can be obtained by solving an optimal set-covering problem. The challenge is in the sizes of the corresponding set-covering problems and in how these can be tamed. The set-covering perspective on symmetry breaking opens up a range of new opportunities deriving from decades of studies on both precise and approximate techniques for this problem. Application of our approach leads to optimal LexLeader symmetry breaks for graphs of order $n\leq 10$ as well as to partial symmetry breaks which improve on the state-of-the-art.