The replica symmetric phase of random constraint satisfaction problems

TL;DR

This paper rigorously confirms physics-based predictions about phase transitions in random constraint satisfaction problems, including thresholds like condensation and reconstruction, and explores implications for Bayesian inference.

Contribution

It proves the predicted phase transition thresholds for a broad class of problems and derives contiguity results relevant to Bayesian inference tasks.

Findings

Confirmed physics predictions on phase transition thresholds

Identified the condensation and reconstruction thresholds

Established implications for Bayesian inference

Abstract

Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution [Krzakala et al., PNAS 2007] physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g., [Banks et al., COLT 2016]).