The large deviations of the whitening process in random constraint satisfaction problems

TL;DR

This paper investigates the large deviations in the whitening process of random constraint satisfaction problems, identifying phase transitions related to frozen variables and solutions' structure, with implications for understanding solution space geometry and algorithmic performance.

Contribution

It refines the understanding of freezing transitions by estimating the disappearance of unfrozen solutions and characterizes atypical solutions through large deviation analysis of the whitening process.

Findings

Identification of the freezing transition where unfrozen solutions vanish.

Discovery of phase transitions for locked solutions with all variables frozen.

Relevance to algorithmic performance by linking solution types to heuristic outputs.

Abstract

Random constraint satisfaction problems undergo several phase transitions as the ratio between the number of constraints and the number of variables is varied. When this ratio exceeds the satisfiability threshold no more solutions exist; the satisfiable phase, for less constrained problems, is itself divided in an unclustered regime and a clustered one. In the latter solutions are grouped in clusters of nearby solutions separated in configuration space from solutions of other clusters. In addition the rigidity transition signals the appearance of so-called frozen variables in typical solutions: beyond this threshold most solutions belong to clusters with an extensive number of variables taking the same values in all solutions of the cluster. In this paper we refine the description of this phenomenon by estimating the location of the freezing transition, corresponding to the disappearance of all unfrozen solutions (not only typical ones). We also unveil phase transitions for the existence and uniqueness of locked solutions, in which all variables are frozen. From a technical point of view we characterize atypical solutions with a number of frozen variables different from the typical value via a large deviation study of the dynamics of a stripping process (whitening) that unveils the frozen variables of a solution, building upon recent works on atypical trajectories of the bootstrap percolation dynamics. Our results also bear some relevance from an algorithmic perspective, previous numerical studies having shown that heuristic algorithms of various kinds usually output unfrozen solutions.