Phase transitions and the search problem
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
A general upper bound for the satisfiability threshold of random r-SAT formulae
Journal of Algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The phase transition in a random hypergraph
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The Asymptotic Order of the Random k -SAT Threshold
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Combinatorial sharpness criterion and phase transition classification for random CSPs
Information and Computation
Random Structures & Algorithms - Proceedings of the Eleventh International Conference "Random Structures and Algorithms," August 9—13, 2003, Poznan, Poland
Survey propagation: An algorithm for satisfiability
Random Structures & Algorithms
Threshold values of random K-SAT from the cavity method
Random Structures & Algorithms
On the solution-space geometry of random constraint satisfaction problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Relaxed survey propagation for the weighted maximum satisfiability problem
Journal of Artificial Intelligence Research
The decimation process in random k-SAT
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
The condensation transition in random hypergraph 2-coloring
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Catching the k-NAESAT threshold
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is 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=12, but they do not 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). Our 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.