Polarised random k-SAT

TL;DR

This paper introduces polarized random k-SAT, a variation of the classic problem, and shows that its satisfiability threshold provides an upper bound for the original model, with a conjecture of asymptotic equality.

Contribution

It defines polarized random k-SAT and proves that its satisfiability threshold is an upper bound for the classical model, proposing they may coincide asymptotically.

Findings

Threshold does not decrease as polarization parameter p moves away from 1/2.

Satisfiability threshold for polarized model bounds the classical model's threshold from above.

Conjecture that the thresholds asymptotically coincide.

Abstract

In this paper we study a variation of the random $k$-SAT problem, called polarized random $k$-SAT. In this model there is a polarization 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 polarized 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 $p=0$ 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 polarized random $k$-SAT is an upper bound on the threshold for random $k$-SAT. In fact, we conjecture that asymptotically the two thresholds coincide.