Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random @math -SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, beyond intuition, there has been scant rigorous evidence that “practical” algorithms are affected by these phase transitions. In this paper we prove that Walksat, a popular randomized satisfiability algorithm, fails on random @math -SAT formulas not very far above clause variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters. Specifically, we prove that Walksat is ineffective with high probability (w.h.p.) if @math , where @math is the number of clauses, @math is the number of variables, and @math is an absolute constant. By comparison, Walksat is known to find satisfying assignments in linear time w.h.p. if @math [A. Coja-Oghlan and A. Frieze, SIAM J. Comput., 43 (2...
Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that Φ is satisfiable for clause variable ratios m n ≤ rk 2k ln 2 with high probability (Achlioptas, Moore: SICOMP 2006; Achlioptas, Peres: J. AMS 2004). Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m n rk · ln(k) k with a non-vanishing probability. In fact, the density m n rk · ln(k) k seems to form a barrier for a broad class of local search algorithms (Achlioptas, Coja-Oghlan: FOCS 2008). On the basis of deep but non-rigorous statistical mechanics considerations, a message passing algorithm called belief propagation guided decimation for solving random k-SAT has been forward (Mezard, Parisi, Zecchina: Science 2002; Braunstein, Mezard, Zecchina: RSA 2005). Experiments suggest that the algorithm might succeed for densities very close to rk for k = 3, 4, 5 (Kroc, Sabharwal, Selman: SAC 2009). Furnishing the first rigorous analysis of belief propagation guided decimation on random k-SAT, the present paper shows that the algorithm fails to find a satisfying assignment already for m n ≥ ρ · rk k, for a constant ρ > 0 independent of k. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math .