On the satisfiability of random $3$-SAT formulas with $k$-wise independent clauses

TL;DR

This paper explores the satisfiability of random 3-SAT formulas generated under complex distributions that exhibit k-wise independence, extending classical models that assume full independence among clauses.

Contribution

It introduces a framework for analyzing satisfiability thresholds in 3-SAT formulas with k-wise independent clause distributions, broadening understanding beyond traditional independent clause models.

Findings

Analyzes satisfiability under k-wise independent clause distributions.

Provides conditions for satisfiability and unsatisfiability in these models.

Extends classical random 3-SAT results to richer distribution families.

Abstract

The problem of identifying the satisfiability threshold of random $3$-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of $n$ Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters $n$, $m$, and $k$, we denote by $\DistFamily_k(n,m)$ the family of probability distributions that produce formulas with $m$ clauses, each selected uniformly at random from all sets of three literals from the $n$ variables, so that the clauses are $k$-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in $\DistFamily_k(n,m)$ for different values of the parameters $n$, $m$, and $k$.