Frozen variables in random boolean constraint satisfaction problems

TL;DR

This paper precisely determines the freezing threshold in certain random boolean constraint satisfaction problems, revealing a phase transition in the structure of solutions related to the density of constraints.

Contribution

It provides the exact freezing threshold for models like NAE-SAT and hypergraph 2-colouring, linking it to the clustering phenomenon and algorithmic complexity.

Findings

For densities above r^f, most solutions have linearly many frozen variables.

Below r^f, solutions typically have only o(n) frozen variables.

The freezing threshold acts as an algorithmic barrier for solving these problems.

Abstract

We determine the exact freezing threshold, r^f, for a family of models of random boolean constraint satisfaction problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large. If the constraint-density of a random CSP, F, in our family is greater than r^f then for almost every solution of F, a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraint-density is less than r^f, then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by non-rigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: that for densities below r^f the random CSPs can be solved using very simple algorithms, while for densities above r^f one requires more sophisticated techniques in order to deal with frozen clusters.