Breaking Symmetries with Involutions

TL;DR

This paper explores using involution-derived graph patterns to create efficient, strong symmetry-breaking constraints for graphs, improving over existing partial symmetry-breaking methods.

Contribution

It introduces a novel approach leveraging involution-based graph patterns to construct concise and effective symmetry-breaking constraints.

Findings

Patterns from involutions help identify large sets of non-canonical graphs.

The proposed constraints are small yet break many symmetries.

Results show improved symmetry breaking over existing partial methods.

Abstract

Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered too weak to be of practical use. Recently, the concept of graph patterns has been introduced and provides a concise representation for (large) sets of non-canonical graphs, i.e.\ graphs that are not lex-leaders and can be excluded from search. In particular, four (specific) graph patterns apply to identify about 3/4 of the set of all non-canonical graphs. Taking this approach further we discover that graph patterns that derive from permutations that are involutions play an important role in the construction of symmetry breaks for graphs. We take advantage of this to guide the construction of partial and complete symmetry breaking constraints based on graph patterns. The resulting constraints are small in size and strong in the number of symmetries they break.