Relaxation and Metastability in the RandomWalkSAT search procedure

TL;DR

This paper analyzes the behavior of the WalkSAT local search algorithm on random Boolean satisfiability problems, revealing a phase transition from linear to exponential resolution times related to metastability and barrier crossing.

Contribution

It introduces a quantum-based expansion scheme to compute relaxation times and proposes an annealed calculation for barrier heights, clarifying the nature of the phase transition.

Findings

Resolution time grows linearly below alpha_d and exponentially above alpha_d.

Metastable states trap the system, causing exponential resolution times.

The phase transition is not linked to solution clustering.

Abstract

An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.