112 research outputs found
Computational Complexity and Phase Transitions
Phase transitions in combinatorial problems have recently been shown to be
useful in locating "hard" instances of combinatorial problems. The connection
between computational complexity and the existence of phase transitions has
been addressed in Statistical Mechanics and Artificial Intelligence, but not
studied rigorously.
We take a step in this direction by investigating the existence of sharp
thresholds for the class of generalized satisfiability problems defined by
Schaefer. In the case when all constraints are clauses we give a complete
characterization of such problems that have a sharp threshold.
While NP-completeness does not imply (even in this restricted case) the
existence of a sharp threshold, it "almost implies" this, since clausal
generalized satisfiability problems that lack a sharp threshold are either
1. polynomial time solvable, or
2. predicted, with success probability lower bounded by some positive
constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE
Conference on Computational Complexit
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