2005/06/30 by Hervé Daudé, Marc Mézard, Marc Mezard +2 · 2 citations
Computer Science · Physics and Astronomy · #Bayesian Modeling and Causal Inference #Constraint Satisfaction and Optimization #Data Management and Algorithms #cond-mat.dis-nn #cs.CC
paper · pdf · doi:10.1016/j.tcs.2008.01.005
published as Theoretical Computer Science 393 (2008) 260-279
arxiv created 2007/09/19 · openalex publication_date 2008/02/28 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/02
We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable if there exist two SAT assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=1/2, but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.