Polarised random -SAT

Abstract

Abstract In this paper we study a variation of the random $k$ -SAT problem, called polarised random $k$ -SAT, which contains both the classical random $k$ -SAT model and the random version of monotone $k$ -SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter $p$ , and in half of the clauses each variable occurs negated with probability $p$ and pure otherwise, while in the other half the probabilities are interchanged. For $p=1/2$ we get the classical random $k$ -SAT model, and at the other extreme we have the fully polarised model where $p=0$ , or 1. Here there are only two types of clauses: clauses where all $k$ variables occur pure, and clauses where all $k$ variables occur negated. That is, for $p=0$ , and $p=1$ , we get an instance of random monotone $k$ -SAT. We show that the threshold of satisfiability does not decrease as $p$ moves away from $\frac{1}{2}$ and thus that the satisfiability threshold for polarised random $k$ -SAT with $p\neq \frac{1}{2}$ is an upper bound on the threshold for random $k$ -SAT. Hence the satisfiability threshold for random monotone $k$ -SAT is at least as large as for random $k$ -SAT, and we conjecture that asymptotically, for a fixed $k$ , the two thresholds coincide.