Convergence of Warning Propagation Algorithms for Random Satisfiable Instances

Abstract

Message propagation algorithms are very effective in finding satisfying assignments for random kSAT instances and hard regions become more narrow.Unfortunately,this phenomenon is still lacks rigorous theoretical proofs.The Warning Propagation(WP) algorithm is the most basic message propagation algorithm.In order to analysis the WP algorithm convergence for random kCNF formulas,the study gives the sharp threshold point for the existence of cycles in the factor graph of random kCNF formulas,the threshold for the existence of cycles in model G(n,k,p) of random kCNF formulas is p=1/8n2 for k=3,p=d/n2.When d becomes asymptotically equal to 1/8,cycles begin to appear,but each component contains at most one cycle,the number of the components containing a single cycle and the length of cycle are a constant independent of n.Thus,the factor graph consists of a forest of trees plus a few components that have a single cycle.Then WHP(with high probability) after at most O(logn+s) iterations,WP converges on these instances.Here s is the size of the connected component.