A better upper bound for the unsatisfiability threshold

Abstract

Let be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number such that if the ratio of the number of clauses over the number of variables of strictly exceeds , then is almost certainly unsatissable. By a well known and more or less straightforward argument, it can be shown that 5:191. This upper bound was improved by Kamath, Motwani, Palem, and Spirakis to 4.758, by rst providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of is around 4.2. In this work, we show that this upper bound can be improved to 4.667. Our proof is elementary and short, and does not use unveriiable mechanical calculations. Moreover it generalizes in a straightforward manner to k-SAT, for k > 3.