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

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 [Formula presented] of constraints per variable, reaching a solution takes a time [Formula presented] growing linearly [Formula presented] or exponentially [Formula presented] with the size N of the instance. The relaxation time [Formula presented] in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For [Formula presented] 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 [Formula presented] of this barrier is proposed. The polynomial to exponential cross-over [Formula presented] is not related to the onset of clustering among solutions occurring at [Formula presented].