Combinatorial Landscape Analysis for k-SAT Instances

Abstract

Over the past ten years, methods from statistical physics have provided a deeper inside into the average complexity of hard combinatorial problems, culminating in a rigorous proof for the asymptotic behaviour of the k-SAT phase transition threshold by Achlioptas and Peres in 2004. On the other hand, when dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase might be helpful for an appropriate adjustment of local search-based procedures. In the present paper, we address both issues in the context of landscapes induced by k-SAT instances: Firstly, we utilize a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. Secondly, we outline a method for obtaining upper bounds for the average number of local maxima in k-SAT instances which indicates some kind of phase transition for the neighbourhood-specific ratio m/n = Theta(2k/k).