# Satisfiability Threshold for Power Law Random 2-SAT in Configuration Model

## TL;DR

This paper investigates the satisfiability threshold in power-law distributed random 2-SAT problems within the configuration model, revealing that the threshold depends on the moments of the degree distribution.

## Contribution

It introduces a detailed analysis of the satisfiability threshold for non-uniform, power-law degree distributions in the configuration model for 2-SAT.

## Key findings

- A satisfiability threshold exists for power-law distributions.
- The threshold is determined by the relation between the first and second moments of the degree distribution.
- The threshold behavior varies with the parameters of the power-law tail.

## Abstract

The Random Satisfiability problem has been intensively studied for decades. For a number of reasons the focus of this study has mostly been on the model, in which instances are sampled uniformly at random from a set of formulas satisfying some clear conditions, such as fixed density or the probability of a clause to occur. However, some non-uniform distributions are also of considerable interest. In this paper we consider Random 2-SAT problems, in which instances are sampled from a wide range of non-uniform distributions. The model of random SAT we choose is the so-called configuration model, given by a distribution $\xi$ for the degree (or the number of occurrences) of each variable. Then to generate a formula the degree of each variable is sampled from $\xi$, generating several \emph{clones} of the variable. Then 2-clauses are created by choosing a random paritioning into 2-element sets on the set of clones and assigning the polarity of literals at random. Here we consider the random 2-SAT problem in the configuration model for power-law-like distributions $\xi$. More precisely, we assume that $\xi$ is such that its right tail $F_{\xi}(x)$ satisfies the conditions $W\ell^{-\alpha}\le F_{\xi}(\ell)\le V\ell^{-\alpha}$ for some constants $V,W$. The main goal is to study the satisfiability threshold phenomenon depending on the parameters $\alpha,V,W$. We show that a satisfiability threshold exists and is determined by a simple relation between the first and second moments of $\xi$.