Solving Non-Boolean Satisfiability Problems with Stochastic Local Search: A Comparison of Encodings

Abstract

Much excitement has been generated by the success of stochastic local search procedures at finding solutions to large, very hard satisfiability problems. Many of the problems on which these procedures have been effective are non-Boolean in that they are most naturally formulated in terms of variables with domain sizes greater than two. Approaches to solving non-Boolean satisfiability problems fall into two categories. In the direct approach, the problem is tackled by an algorithm for non-Boolean problems. In the transformation approach, the non-Boolean problem is reformulated as an equivalent Boolean problem and then a Boolean solver is used. This paper compares four methods for solving non-Boolean problems: one direct and three transformational. The comparison first examines the search spaces confronted by the four methods, and then tests their ability to solve random formulas, the round-robin sports scheduling problem, and the quasigroup completion problem. The experiments show that the relative performance of the methods depends on the domain size of the problem and that the direct method scales better as domain size increases. Along the route to performing these comparisons we make three other contributions. First, we generalize Walksat, a highly successful stochastic local search procedure for Boolean satisfiability problems, to work on problems with domains of any finite size. Second, we introduce a new method for transforming non-Boolean problems to Boolean problems and improve on an existing transformation. Third, we identify sufficient conditions for omitting at-least-one and at-most-one clauses from a transformed formula. Fourth, for use in our experiments we propose a model for generating random formulas that vary in domain size but are similar in other respects.