Sharpness of the satisfiability threshold for non-uniform random K-SAT

Abstract

We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p1, . . ., pn) on the variable occurrences. Therefore, we call it nonuniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a. a. s.) satisfiable to a. a. s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We identify conditions on the variable probability distribution (p1, . . ., pn) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut's sharpness result from uniform to non-uniform random k-SAT and implies sharpness for thresholds of a wide range of random k-SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.