Criticality and Heterogeneity in the Solution Space of Random Constraint Satisfaction Problems

TL;DR

This paper investigates the transition in the solution space of random constraint satisfaction problems, revealing a homogeneity-breaking phase before ergodicity-breaking, with implications for understanding glassy dynamics.

Contribution

It introduces the concept of a homogeneity-breaking transition in the solution space of random CSPs, preceding the known ergodicity-breaking transition, and characterizes the formation of solution communities.

Findings

Solution communities form at a critical constraint density alpha_cm.

The solution space becomes critical at alpha_cm.

Connection to dynamical heterogeneity in lattice glass models.

Abstract

Random constraint satisfaction problems are interesting model systems for spin-glasses and glassy dynamics studies. As the constraint density of such a system reaches certain threshold value, its solution space may split into extremely many clusters. In this paper we argue that this ergodicity-breaking transition is preceded by a homogeneity-breaking transition. For random K-SAT and K-XORSAT, we show that many solution communities start to form in the solution space as the constraint density reaches a critical value alpha_cm, with each community containing a set of solutions that are more similar with each other than with the outsider solutions. At alpha_cm the solution space is in a critical state. The connection of these results to the onset of dynamical heterogeneity in lattice glass models is discussed.