The Satisfiability Threshold Conjecture: Techniques Behind Upper Bound Improvements

Abstract

One of the most challenging problems in probability and complexity theory concerns the establishment and the determination of the satisfiability threshold for random Boolean formulas consisting of clauses with exactly k literals, or k-SAT formulas with emphasis on the case k = 3, or 3-SAT. According to many experimental observations, there exists a critical value rk of the number of clauses to the number of variables ratio r = m/n such that almost all randomly generated formulas with r > rk are unsatisfiable while almost all randomly generated formulas with r < rk are satisfiable. The statement that such a crossover point really exists is called the “satisfiability threshold conjecture”. While experiments hint at such a direction, as far as theoretical work is concerned, progress has been difficult. Up to now, there are rigorous proofs of only successively better upper and lower bounds for the value of the (conjectured) threshold although, in an important advance, Friedgut proved that the phase transition is sharp (without showing the existence of a fixed transition point). In this work, our goal is to review the series of improvements of the upper bounds for 3-SAT and the techniques from which the improvements resulted. We give only a passing reference to the improvements of the lower bounds, as they rely on significantly different techniques that would require much more space to present.