Entropy landscape and non-Gibbs solutions in constraint satisfaction problems

TL;DR

This paper explores the entropy landscape of the bicoloring problem in random graphs, classifies solution clusters, and analyzes how various algorithms navigate these clusters, including their ability to find non-Gibbs solutions beyond phase transitions.

Contribution

It combines the cavity method with algorithm analysis to map solution clusters and identify algorithms capable of finding non-Gibbs solutions in complex constraint satisfaction problems.

Findings

Identified phase transitions in the solution landscape.

Demonstrated that a smoothed decimation algorithm finds solutions in sub-dominant clusters.

Showed non-Gibbs solutions can be located beyond the rigidity transition.

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 type 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 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 sub-dominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen. These non-equilibrium solutions belong to the most probable unfrozen clusters.