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

Florent Krzakala 1, Andrea Montanari 2, Federico Ricci-Tersenghi, Guilhem Semerjian 2, Lenka Zdeborova 3

Proceeding of the national academy of sciences 104, 25 (2007) 10318

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.

  • 1. Laboratoire de Physico-Chimie Théorique (LPCT),
    CNRS : UMR7083 – ESPCI ParisTech
  • 2. Laboratoire de Physique Théorique de l'ENS (LPTENS),
    CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
  • 3. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud