The threshold for random k-SAT is 2k (ln 2 - O(k))

Authors:
Dimitris Achlioptas;Yuval Peres
Affiliations:
Microsoft Research, Redmond, WA;University of California, Berkeley, Berkeley, CA
Venue:
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Year:
2003

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Abstract

Let Fk(n,m) be a random k-SAT formula on n variables formed by selecting uniformly and independently m out of all possible k-clauses. It is well-known that for r ≥ 2k ln 2, Fk(n,rn) is unsatisfiable with probability 1-o(1). We prove that there exists a sequence tk = O(k) such that for r ≥ 2k ln 2 - tk, Fk(n,rn) is satisfiable with probability 1-o(1).Our technique yields an explicit lower bound for every k which for k 3 improves upon all previously known bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.