An instance of a random constraint satisfaction problem defines a random subset 𝒮 (the set of solutions) of a large product spaceXN(the set of assignments). We consider two prototypical problem ensembles (randomk-satisfiability andq-coloring of random regular graphs) and study the uniform measure with support onS. 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 thenlargest states follows a Poisson-Dirichlet process. For typical large instances, the two transitions are sharp. We determine 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 also beat this threshold.

Krz̧akała, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., & Zdeborová, L. (2007). Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25), 10318–10323.