On the freezing of variables in random constraint satisfaction problems

TL;DR

This paper investigates the critical behavior of variable freezing in random constraint satisfaction problems, revealing phase transitions and structural changes in solution spaces, with applications to satisfiability and graph coloring.

Contribution

It develops a formalism to analyze the freezing transition in generic CSPs and connects it to percolation and information theory, extending understanding of solution space structure.

Findings

Identification of the freezing transition as a distinct phase change

Connection between freezing phenomena and percolation models

Application of the formalism to satisfiability and graph coloring problems

Abstract

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.