1203.5521 (Florent Krzakala et al.)

Reweighted belief propagation and quiet planting for random K-SAT    [PDF]

Florent Krzakala, Marc Mézard, Lenka Zdeborová

We study the random K-satisfiability problem using a partition function where each solution is reweighed according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighed partition function. This allows us to obtain several new results on the properties of random K-satisfiability problem. In particular the reweighing allows to introduce a planted ensemble that generates instances that are, in some region of parameters, equivalent to random instances. We are hence able to generate at the same time a typical SAT instance and one of its solutions. We study the relation between clustering and belief propagation fixed points and we give a direct evidence for the existence of purely entropic (rather than energetic) barriers between clusters in some region of parameters in the random K-satisfiability problem. We show explicitly how to find solutions of random K-SAT leading to a non-trivial whitening core; such solutions were known to exist but were so far never found on large instances. Finally, we discuss algorithmic hardness of such planted instances and determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.

View original: http://arxiv.org/abs/1203.5521