Generating hard satisfiability problems
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
Artificial Intelligence
A threshold for unsatisfiability
Journal of Computer and System Sciences
Typical random 3-SAT formulae and the satisfiability threshold
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Mick gets some (the odds are on his side) (satisfiability)
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
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We investigate phase transitions for the family of bounded satisfiability problems 3SAT(b), recently introduced by Zhang, that ask: given a 3CNF- formula, is there a truth assignment that violates no more than b of its clauses. Zhang's results were experimental and for a fixed number of variables (n = 25), and suggested that the locations of the phase transitions for 3SAT(b) are separated and move significantly as b increases. Analysis of these locations was posed as an open question. We analytically show that the phase transitions of all 3SAT(6) problems must occur within a narrow region, regardless of how large the value of b is. Moreover, our experiments reveal that the phase transitions for these problems occur in a remarkable way. Specifically, unlike 3SAT, the probability curves for 3SAT(6) do not have a quasi-common intersection point about which they rotate as they become steeper with increasing n. Instead, they move rapidly to the left toward the narrow region that the analysis predicts.