On belief propagation guided decimation for random k-SAT

TL;DR

This paper rigorously analyzes the Belief Propagation Guided Decimation algorithm for random k-SAT problems, showing it fails to find solutions beyond certain clause-to-variable ratios, thus clarifying its limitations.

Contribution

It provides the first rigorous analysis of BP decimation, establishing its failure at densities above a specific threshold, contrasting previous non-rigorous predictions.

Findings

BP decimation fails for m/n > c * r(k)/k

Theoretical bounds on the algorithm's success threshold

Clarifies limitations of message passing algorithms for k-SAT

Abstract

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).