Reweighted belief propagation and quiet planting for random K-SAT

TL;DR

This paper introduces a reweighted belief propagation approach to analyze random K-SAT, revealing new insights into solution clustering, entropic barriers, and generating challenging planted instances with known solutions.

Contribution

It develops a novel reweighted partition function framework and applies belief propagation to generate and analyze complex random K-SAT instances with known solutions.

Findings

Identification of entropic barriers between solution clusters.

Generation of large planted instances with solutions and high algorithmic hardness.

Evidence of solutions with non-trivial whitening cores in large instances.

Abstract

We study the random K-satisfiability problem using a partition function where each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several new results on the properties of random K-satisfiability problem. In particular the reweighting 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 random 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 exhibit, in some large planted instances, solutions with a non-trivial whitening core; such solutions were known to exist but were so far never found on very large instances. Finally, we discuss algorithmic hardness of such planted instances and we determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.