Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

Abstract

We study the entropy landscape of solutions for the bicoloring problem in random graphs, a representative difficult constraint satisfaction problem. Our goal is to classify which types of clusters of solutions are addressed by different algorithms. In the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem--e.g., dynamical, rigidity, and satisfiability-unsatisfiability (SAT-UNSAT) transitions. In the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem. For instance, we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so-called rigidity transition where the thermodynamically relevant clusters become frozen. These nonequilibrium solutions belong to the most probable unfrozen clusters.

Statistical mechanics of maximal independent sets

“…Our numerical calculations indicate that BP decimation gives the best performances, but the values at which we find solutions are still far from the theoretically predicted bounds. Notice that despite the dynamical transition in the low density region, the absence of globally frozen variables could make the problem easy on average in that region [46,51].…”

“…In the large deviations cavity formalism, it is possible to compute the number of solutions of a CSP at a distance d from a given one σ * using the weight enumerator function [45,50,51] where s(d) is the entropy of solutions at a distance d =…”

Exhaustive enumeration unveils clustering and freezing in the random 3-satisfiability problem

“…The first is a quantitative comparison between the number of clusters of solutions (glassy states) and its analytical prediction [9,15,16,17]. The second is the location of the freezing transition which was recently suggested to be responsible for computational hardness of the random satisfiability problem [14,18,19], but not yet computed in the 3-SAT problem.Clustering and freezing -In physics of glassy systems, clusters correspond to pure thermodynamical states and are being described in the literature about glasses and spin glasses for more than one quarter of a century [7]. A formal definition of clusters in K-SAT as extremal Gibbs measures was given recently in [10].…”

“…The first is a quantitative comparison between the number of clusters of solutions (glassy states) and its analytical prediction [9,15,16,17]. The second is the location of the freezing transition which was recently suggested to be responsible for computational hardness of the random satisfiability problem [14,18,19], but not yet computed in the 3-SAT problem.…”

“…Several theoretical investigations of a related rigidity transition, where clusters which contain almost all the solution become frozen, can be found in [13,14,26]. But as long as soft clusters exist some algorithms may be able to find them, as shown in [19]. A related numerical study [22] investigates the size dependence of the fraction of frozen solutions at α = 4.20 < α f .…”