Quiet Planting in the Locked Constraint Satisfaction Problems

TL;DR

This paper investigates the structure of locked constraint satisfaction problems, identifying the hard region where instances typically have a unique solution, using advanced statistical physics methods and probabilistic analysis.

Contribution

It establishes the connection between random and planted ensembles and locates the hard region in the planted ensemble where solutions are typically unique.

Findings

Identifies the hard region in the planted ensemble.

Shows instances often have a single satisfying assignment in the hard region.

Uses cavity method and reconstruction on trees for analysis.

Abstract

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.