Exploiting Isomorphic Subgraphs in SAT

Abstract

While static symmetry breaking has been explored in the SAT community for decades, only as of 2010 research has focused on exploiting the same discovered symmetry dynamically, during the run of the SAT solver, by learning extra clauses. The two methods are distinct and not compatible. The former may prune solutions, whereas the latter does not – it only prunes areas of the search that are guaranteed not to have solutions, like standard conflict clauses. Both approaches, however, require what we call full symmetry, namely a propositionally-consistent mapping σ between the literals, such that σ(φ) ≡ φ, where here ≡ means syntactic equivalence modulo clause ordering and literal ordering within the clauses. In this article we show that such full symmetry is not a necessary condition for adding extra clauses: isomorphism between possibly-overlapping subgraphs of the colored incidence graph is sufficient. While finding such subgraphs is a computationally hard problem, there are many cases in which they can be detected a priori by analyzing the high-level structure of the problem from which the CNF was derived. We demonstrate this principle with several well-known problems.