Inhomogeneous random 2-SAT

TL;DR

This paper introduces an inhomogeneous random 2-SAT model where variables have types influencing clause formation, and identifies a spectral radius-based threshold that determines satisfiability with high probability.

Contribution

It extends the classical 2-SAT threshold result to inhomogeneous models using a spectral radius criterion derived from a kernel function.

Findings

Spectral radius $ ho^*(W)$ determines satisfiability threshold.

$ ho^*(W)<1$ implies almost sure satisfiability.

$ ho^*(W)>1$ implies almost sure unsatisfiability.

Abstract

We introduce an inhomogeneous variant of random 2-SAT. Each variable $v_1,\ldots,v_n$ is assigned a type from a state space $\Lambda$, independently at random. Clause inclusion is governed by a symmetric measurable kernel $W$ on $(\Lambda \times \{+,-\})^2$, in analogy with the inhomogeneous random graph model of Bollob\'as, Janson, and Riordan: given literals $\ell_i\in\{v_i,\neg v_i\}$ and $\ell_j\in\{v_j,\neg v_j\}$, the clause $\{\ell_i,\ell_j\}$ appears with probability $W(\mathrm{type}(\ell_i),\mathrm{type}(\ell_j))/(2n)$. In particular, for a variable $v_i$ of type $x\in\Lambda$, the slices $W((+,x),\cdot)$ and $W((-,x),\cdot)$ describe how $v_i$ and $\neg v_i$ interact with other literals. We identify a parameter $\rho^*(W)$, defined as the spectral radius of an integral operator derived from $W$, and show that $\rho^*(W)<1$ and $\rho^*(W)>1$ correspond to asymptotically almost surely satisfiable and unsatisfiable instances, respectively. The satisfiability threshold for homogeneous random 2-SAT is well-established, occurring when the ratio of clauses to variables is $1$. This corresponds to a weight function of $W \equiv 1$ and a clause density of $1/(2n)$. Our result extends this classical result to a broad class of models controlled by types of variables.