On Belief Propagation Guided Decimation for Random k-SAT

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 ∼ 2 ln 2 with high probability. 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. 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 rk. 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 ≥ ρ · rk/k, for a constant ρ > 0 independent of k.