Relaxation and metastability in a local search procedure for the random satisfiability problem

Abstract

An analysis of the average properties of a local search procedure (RandomWalkSAT) for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, reaching a solution takes a time T(res) growing linearly [T(res) approximately tau(res)(alpha)N, alpha<alpha(d)] or exponentially (T(res) approximately exp[N zeta(alpha)], alpha>alpha(d)) with the size N of the instance. The relaxation time tau(res)(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 to exponential cross-over alpha(d) approximately =2.7 is not related to the onset of clustering among solutions occurring at alpha approximately =3.86.