Local symmetry breaking in the satisfiability problem

Abstract

Abstract. The SAT problem is shown to be the first decision NPcomplete problem (Cook,71). It is central in complexity theory. In the last decade, the satisfiability proof procedures are improved by symmetry elimination. A CNF formula usually contains an interesting number of symmetries. There are two kinds of symmetry exploitation. The first one corresponds to global symmetry breaking, that is, only the symmetries of the initial problem (the problem at the root of the search tree) are detected and eliminated. The second one deals with all local symmetries that appear at each node of the search tree. Local symmetry has to be detected and eliminated dynamically during the search. Exploiting such symmetries seems to be a hard task. Almost all of the known works on symmetry in satisfiability are on global symmetry. Only few works are carried on local symmetry, despite their importance in practice. An important challenge is then to detect and break local symmetries efficiently during the search. The work that we present here is a contribution towards an answer to this hard challenge. We present a new method for local symmetries breaking that consists in detecting dynamically local symmetries by reducing the remaining partial SAT instance at each node of the search tree to a graph that has an equivalent automorphism group than the symmetry group of the partial SAT instance. We used the software Saucy to compute the automorphism group and implemented a local symmetry cut in a SAT solver. We experimented this method on several SAT instances and compared it with a method exploiting global symmetries. The results obtained are very promising. Local symmetry improves global symmetry on some hard instances and is complementary to global symmetry.