Clustering of solutions in hard satisfiability problems

Abstract

We study numerically the solution space structure of random 3-SAT problemsclose to the SAT/UNSAT transition. This is done by considering chains ofsatisfiability problems, where clauses are added sequentially to a problem instance.Using the overlap measure of similarity between different solutions found onthe same problem instance, we examine geometrical changes as a function ofα. In each chain, the overlap distribution is first smooth, but then develops a tiered structure,indicating that the solutions are found in well separated clusters. On chains of not too largeinstances, all remaining solutions are eventually observed to be found in only one small clusterbefore vanishing. This condensation transition point is estimated by finite size scaling to beαc = 4.26 with an apparent critical exponent of about1.7. The average overlap value is also observed to increase withα up to the transition, indicating a reduction in solutions space size, in accordancewith theoretical predictions. The solutions are generated by a local heuristic,ASAT, and compared to those found by the Survey Propagation algorithm up toαc.