Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Randomized algorithms
Asymptotic enumeration methods
Handbook of combinatorics (vol. 2)
Approximating the unsatisfiability threshold of random formulas
Random Structures & Algorithms
Typical random 3-SAT formulae and the satisfiability threshold
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Bounding the unsatisfiability threshold of random 3-SAT
Random Structures & Algorithms
Optimal myopic algorithms for random 3-SAT
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
A spectral technique for random satisfiable 3CNF formulas
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Solving random satisfiable 3CNF formulas in expected polynomial time
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Phase transitions of PP-complete satisfiability problems
Discrete Applied Mathematics
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The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the (experimentally observed) abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. An upper bound of 4.506 was announced by Dubois et al. in 1999 but, to the best of our knowledge, no complete proof has been made available from the authors yet. We consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we explain how the method of local maximum satisfying truth assignments can be combined withresu lts for coupon collector's probabilities in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. Thus, we improve over the best, with an available complete proof, previous upper bound, which was 4.596. In order to obtain this value, we also establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients) which may be of independent interest.