Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems

TL;DR

This paper analyzes the phase transitions in random constraint satisfaction problems, identifying the precise locations of clustering and condensation points, and relates these phases to the effectiveness of various algorithms.

Contribution

It provides the first precise determination of the phase transition points and formal definitions based on variable correlations in random CSPs.

Findings

Clustering occurs at a specific constraint density, fragmenting solutions into pure states.

Condensation occurs at a higher constraint density, with solutions concentrating on a few dominant states.

Local algorithms are effective up to the clustering phase, while advanced message passing can perform beyond that.

Abstract

An instance of a random constraint satisfaction problem defines a random subset S (the set of solutions) of a large product space (the set of assignments). We consider two prototypical problem ensembles (random k-satisfiability and q-coloring of random regular graphs), and study the uniform measure with support on S. As the number of constraints per variable increases, this measure first decomposes into an exponential number of pure states ("clusters"), and subsequently condensates over the largest such states. Above the condensation point, the mass carried by the n largest states follows a Poisson-Dirichlet process. For typical large instances, the two transitions are sharp. We determine for the first time their precise location. Further, we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem. The degree of correlation naturally affects the performances of many search/sampling algorithms. Empirical evidence suggests that local Monte Carlo Markov Chain strategies are effective up to the clustering phase transition, and belief propagation up to the condensation point. Finally, refined message passing techniques (such as survey propagation) may beat also this threshold.