# Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT

## TL;DR

This paper establishes bounds on the satisfiability threshold for random k-SAT instances with power law variable distributions, revealing a phase transition at a specific exponent value and extending understanding beyond uniform models.

## Contribution

It introduces a rigorous analysis of random k-SAT with power law variable distributions, identifying a precise satisfiability threshold based on the distribution's exponent.

## Key findings

- Threshold at (2k-1)/(k-1)
- Instances are unsatisfiable w.h.p. below the threshold
- Instances are satisfiable w.h.p. above the threshold

## Abstract

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, $m=\Theta(n)$ clauses, and a power law distribution on the variable occurrences with exponent $\beta$. We observe a satisfiability threshold at $\beta=(2k-1)/(k-1)$. This threshold is tight in the sense that instances with $\beta\le(2k-1)/(k-1)-\varepsilon$ for any constant $\varepsilon>0$ are unsatisfiable with high probability (w.h.p.). For $\beta\geq(2k-1)/(k-1)+\varepsilon$, the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios $m/n$; they are unsatisfiable above a ratio $m/n$ that depends on $\beta$.