Satisfying assignments of Random Boolean CSP: Clusters and Overlaps

The distribution of overlaps of solutions of a random 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 confirmed in \cite{cond-mat/0504070/prl} \cite{cond-mat/0506053}. There it is proved that the overlap distribution of random $k$-SAT, $k\geq 9$, has discontinuous support. Furthermore, Achlioptas and Ricci-Tersenghi proved that, for random $k$-SAT, $k\geq 8$. and constraint densities close enough to the phase transition there exists an exponential number of clusters of satisfying assignments; moreover, 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 (STOC 2002) 1. For all constraint sets $S$ (described explicitly in Creignou and Daude (2004), Istrate (2005)) s.t. $SAT(S)$ has a sharp threshold and all $q\in (0,1]$, $q$-overlap-$SAT(S)$ has a sharp threshold (i.e. the first step of the approach in Mora et al. works in all nontrivial cases). 2. For any constraint density value $c<1$, the set of solutions of a random instance of 2-SAT form, w.h.p., a single cluster. Also, for and any $q\in (0,1]$ such an instance has w.h.p. two satisfying assignment of overlap $\sim q$. Thus, as expected from Statistical Physics predictions, the second step of the approach in Mora et al. fails for 2-SAT.