Analysing Survey Propagation Guided Decimationon Random Formulas

Abstract

Let vec(theta) be a uniformly distributed random k-SAT formula with n variables and m clauses. For clauses/variables ratio m/n <= r_{k-SAT} ~ 2^k*ln(2) the formula vec(theta) is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond m/n ~ 2k*ln(k)/k with a non-vanishing probability. Non-rigorous statistical mechanics work on k-CNF led to the development of a new efficient "message passing algorithm" called Survey Propagation Guided Decimation [Mézard et al., Science 2002]. Experiments conducted for k=3,4,5 suggest that the algorithm finds satisfying assignments close to r_{k-SAT}. However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random k-SAT formulas efficiently already for m/n = 2^{k}(1 + epsilon_k)*ln(k)/k with lim_{k -> infinity} epsilon_k = 0 almost a factor k below r_{k-SAT}.