Random Satisfiability

Satisfiability has received a great deal of study as the canonical NP-complete problem. In the last twenty years a significant amount of this effort has been devoted to the study of randomly generated satisfiability instances and the performance of different algorithms on them. Historically, the motivation for studying random instances has been the desire to understand the hardness of “typical” instances. In fact, some early results suggested that deciding satisfiability is “easy on average”. Unfortunately, while “easy” is easy to interpret, “on average” is not. One of the earliest and most often quoted results for satisfiability being easy on average is due to Goldberg [Gol79]. In [FP83], though, Franco and Paull pointed out that the distribution of instances used in the analysis of [Gol79] is so greatly dominated by “very satisfiable” formulas that if one tries truth assignments completely at random, the expected number of trials until finding a satisfying one is O(1). Alternatively, Franco and Paull pioneered the analysis of random instances of k-SAT, i.e., asking the satisfiability question for random kCNF formulas (defined precisely below). Among other things, they showed [FP83] that for all k ≥ 3 the DPLL algorithm needs an exponential number of steps to report all cylinders of solutions of such a formula, or that no solutions exist.