The distribution of overlaps of solutions of a random constraint satisfaction problem (CSP) is an indicator of the overall geometry of its solution space. For random k-SAT, nonrigorous methods from Statistical Physics support the validity of the one step replica symmetry breaking approach. Some of these predictions were rigorously conf irmed in (Met al. 2005a) (Mezard et al. 2005b). There it is proved that the overlap distribution of random k-SAT, k ≥ 9, has discontinuous support. Furthermore, Achlioptas and Ricci-Tersenghi (Achlioptas and Ricci-Tersenghi 2006) proved that, for random k-SAT, k ≥ 8, and constraint densities close enough to the phase transition: - there exists an exponential number of clusters of satisfying assignments. - the distance between satisfying assignments in different clusters is linear. We aim to understand the structural properties of random CSP that lead to solution clustering. To this end, we prove two results on the cluster structure of solutions for binary CSP under the random model from (Molloy 2002):
['Gabriel Istrate']
Satisfying Assignments of Random Boolean Constraint Satisfaction Problems: Clusters and Overlaps.