On Belief Propagation Guided Decimation for Random k-SAT

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Let F be a uniformly distributed random k-SAT formula with n variables and m clauses. Non-constructive arguments show that F is satisfiable for clause/variable ratios m/n< r(k)~2^k ln 2 with high probability. Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n r(k).ln(k)/k with a non-vanishing probability. In fact, the density m/n r(k).ln(k)/k seems to form a barrier for a broad class of local search algorithms. One of the very few algorithms that plausibly seemed capable of breaking this barrier is a message passing algorithm called Belief Propagation Guided Decimation. It was put forward on the basis of deep but non-rigorous statistical mechanics considerations. Experiments conducted for k=3,4,5 suggested that the algorithm might succeed for densities very close to r_k. Furnishing the first rigorous analysis of BP decimation, the present paper shows that the algorithm fails to find a satisfying assignment already for m/n>c.r(k)/k, for a constant c>0 (independent of k).