# A Model of Random Industrial SAT

## TL;DR

This paper introduces a new model of industrial SAT that incorporates community structure among variables, and studies how this affects the satisfiability threshold, finding it tends to be lower or even vanish compared to standard random SAT.

## Contribution

It proposes a community-structured random SAT model and analyzes its satisfiability threshold, revealing significant differences from traditional random SAT models.

## Key findings

- Threshold in the new model is smaller than in random SAT.
- Under certain conditions, the threshold vanishes.
- Community structure influences satisfiability properties.

## Abstract

One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT instances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a model of random industrial SAT, in which the central generalization with respect to random SAT is the additional community structure. There has been a lot of work on the satisfiability threshold of random $k$-SAT, starting with the calculation of the threshold of $2$-SAT, up to the recent result that the threshold exists for sufficiently large $k$. In this paper, we endeavor to study the satisfiability threshold for the proposed model of random industrial SAT. Our main result is that the threshold in this model tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.