Biased random k-SAT

The basic random $k$-SAT problem is: Given a set of $n$ Boolean variables, and $m$ clauses of size $k$ picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive -- i.e. variables occur negated w.p. $0<p< \frac{1}{2}$ and positive otherwise -- and study how the satisfiability threshold depends on $p$. For $p<\frac{1}{2}$ this model breaks many of the symmetries of the original random $k$-SAT problem, e.g. the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed $k$, we find the asymptotics of the threshold as $p$ approaches $0$ or $\frac{1}{2}$. The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics

Satisfiability threshold of the skewed random k-SAT

Abstract. We consider the satisfiability phase transition in skewed random k-SAT distributions. It is known that the random k-SAT model, in which the instance is a set of m k-clauses selected uniformly from the set of all k-clauses over n variables, has a satisfiability phase transition at a certain clause density. The essential feature of the random k-SAT is that positive and negative literals occur with equal probability in a random formula. How does the phase transition behavior change as the relative probability of positive and negative literals changes? In this paper we focus on a distribution in which positive and negative literals occur with different probability. We present empirical evidence for the satisfiability phase transition for this distribution. We also prove an upper bound on the satisfiability threshold and a linear lower bound on the number of literals in satisfying partial assignments of skewed random k-SAT formulas.