Breaking Symmetries from a Set-Covering Perspective
Michael Codish, Mikoláš Janota
TL;DR
Addresses symmetry in graph search by reframing symmetry breaking as a minimal set-covering problem, where each permutation covers the graphs it reduces and a canonizing set must cover all non-canonical graphs. The authors introduce a concise, pattern-based representation of the cover sets, define backbone permutations that must appear in any optimal cover, and develop symbolic and SAT-based methods to prune candidates and solve the set-cover efficiently. They demonstrate optimal LexLeader-like symmetry breaks for graphs up to $n\le 10$, and show backbone-driven partial breaks can be substantially more precise than transposition-based approaches, while greatly reducing problem size (e.g., a $199\times 197$ remaining instance for $n=10$). The work enables practical, verifiable symmetry breaking with SAT/OPB tooling and opens avenues for pattern- and backbone-driven set-covering in broader graph-search contexts.
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.