Complete Convergence of Message Passing Algorithms for Some Satisfiability Problems

Experimental results show that certain message passing algorithms, namely, Survey Propagation, are very effective in finding satisfying assignments for random satisfiable 3CNF formulas which are considered hard for other SAT heuristics. Unfortunately, rigorous understanding of this phenomena is still lacking. In this paper we make a modest step towards providing rigorous explanation for the effectiveness of message passing algorithms. We analyze the performance of Warning Propagation, a popular message passing algorithm that is simpler than Survey Propagation. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple “off the shelf” heuristic) works when the clause-variable ratio is a sufficiently large constant. We are not aware of previous rigorous analysis of message passing algorithms for satisfiability instances, though such analysis was performed for decoding of Low Density Parity Check (LDPC) Codes. We discuss some of the differences between results for the LDPC setting and our results. The Weizmann Institute. [email protected]. U.C. Berkeley. E-mail: [email protected]. Supported by a Sloan fellowship in Mathematics, by NSF Career award DMS-0548249 and NSF grants DMS-0528488 and DMS-0504245 Tel-Aviv University. E-mail: [email protected].