Branching Process approach for 2-SAT thresholds

TL;DR

This paper extends the analysis of 2-SAT satisfiability thresholds by incorporating clauses with varying numbers of positive literals, using a generalized branching process approach to determine the threshold conditions.

Contribution

It introduces a generalized branching process framework to analyze 2-SAT thresholds with clauses having different positive literal counts, expanding prior models.

Findings

Threshold determined by maximum eigenvalue of branching matrix

Generalizes previous 2-SAT threshold results

Provides a new analytical tool for complex clause distributions

Abstract

It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $\alpha/2n$, exhibits a sharp threshold at $\alpha=1$. We study a more general 2-SAT model in which each clause occurs independently but with probability $\alpha_i/2n$ where $i \in \{0,1,2\}$ is the number of positive literals in that clause. We generalize branching process arguments by Verhoeven(99) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.