Symmetry and Satisfiability: An Update

The past few years have seen significant progress in algorithms and heuristics for both SAT and symmetry detection. Additionally, the thesis that some of SAT’s intractability can be explained by the presence of symmetry, and that it can be addressed by the introduction of symmetry-breaking constraints, was tested, albeit only to a rather limited extent. In this paper we explore further connections between symmetry and satisfiability and demonstrate the existence of intractable SAT instances that exhibit few or no symmetries. Specifically, we describe a highly scalable symmetry detection algorithm based on a decision tree that combines elements of group-theoretic computation and SAT-inspired backtracking search, and provide results of its application on the SAT 2009 competition benchmarks. For SAT instances with significant symmetry we also compare SAT runtimes with and without the addition of symmetry-breaking constraints.