Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2006-12-14
Proc. Natl. Acad. Sci. 104, 10318 (2007)
Physics
Condensed Matter
Statistical Mechanics
6 pages, 6 figures, slightly revised version
Scientific paper
10.1073/pnas.0703685104
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.
Krzakala Florent
Montanari Andrea
Ricci-Tersenghi Federico
Semerjian Guilhem
Zdeborová Lenka
No associations
LandOfFree
Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-476255