Tail bounds for occupancy and the satisfiability threshold conjecture

Abstract

Abstract The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well‐known conjecture on threshold phenomenon for the satisfiability problem. Consider random 3‐SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c * ≈︁ 4.2. We provide a strong upper bound on the value of c *, showing that for c > 4.758 a random 3‐SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.