Approximating the unsatisfiability threshold of random formulas

Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253

[1]  John Franco,et al.  Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem , 1983, Discret. Appl. Math..

[2]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[3]  Bruce A. Reed,et al.  Mick gets some (the odds are on his side) (satisfiability) , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[4]  Alan M. Frieze,et al.  Analysis of Two Simple Heuristics on a Random Instance of k-SAT , 1996, J. Algorithms.

[5]  Yannis C. Stamatiou,et al.  An Inequality for Reducible, Increasing Properties of Randomly Generated Words , 1997 .

[6]  David Aldous.  The harmonic mean formula for probabilities of unions: applications to sparse random graphs , 1989, Discret. Math..

[7]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[8]  David A. Redfern,et al.  The Maple Handbook: Maple V Release 3 , 1995 .

[9]  J. Simon,et al.  Etude statistique de l'existence de solutions de problèmes SAT, application aux systèmes-experts , 1986 .

[10]  Colin McDiarmid.  On a Correlation Inequality of Farr , 1992, Comb. Probab. Comput..

[11]  Yacine Boufkhad,et al.  A General Upper Bound for the Satisfiability Threshold of Random r-SAT Formulae , 1997, J. Algorithms.

[12]  Svante Janson,et al.  Poisson Approximation for Large Deviations , 1990, Random Struct. Algorithms.

[13]  Donald E. Knuth,et al.  fundamental algorithms , 1969 .

[14]  Svante Janson,et al.  New versions of Suen's correlation inequality , 1998, Random Struct. Algorithms.

[15]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[16]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[17]  Stephen Suen,et al.  A correlation inequality and a Poisson limit theorem for nonoverlapping balanced subgraphs of a random graph , 1990, Random Struct. Algorithms.

[18]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .