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

Ioannis Caragiannis, Nick Gravin, Zhile Jiang

TL;DR

The 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 $k$.

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 $k$.

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

TL;DR

The aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in for different values of the parameters , , and .

Abstract

The problem of identifying the satisfiability threshold of random -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 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 , , and , we denote by the family of probability distributions that produce formulas with clauses, each selected uniformly at random from all sets of three literals from the variables, so that the clauses are -wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in for different values of the parameters , , and .

Paper Structure

This paper contains 22 sections, 12 theorems, 46 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Random SAT.
  3. Relaxation of independence.
  4. Problem formulation
  5. Reasonable expectations.
  6. Our results
  7. Lower satisfiability thresholds.
  8. Upper satisfiability thresholds.
  9. Roadmap
  10. Preliminaries
  11. Random $3$-SAT without mutual independence
  12. Tight bounds for the upper satisfiability threshold $\texttt{UST}_2(n)$
  13. A tight lower bound for $\texttt{UST}_3(n)$
  14. An upper bound for $\texttt{UST}_4(n)$
  15. Upper-bounding $\operatorname{\mathbf E}_{C\sim \mathcal{D}} \left [ \kappa(C) \right ]$.
  16. ...and 7 more sections

Key Result

Theorem 3.1

$\texttt{UST}_2(n)=\Theta(n^3)$.

Figures (1)

  • Figure 1: Possible structures formed by four clauses in the corresponding bipartite multi-graph. In subfigure (e), there is one $K_{2,2}$ subgraph formed by nodes $(a,b)$, $(d,g)$, $g$, and $h$ and their incident edges. In subfigure (f), there are two $K_{2,2}$ subgraphs; one formed by the nodes $(a,b)$, $(a,d)$, $g$, and $h$ and their incident edges and one formed by the nodes $(a,g)$, $(a,h)$, $b$, and $d$ and their incident edges.

Theorems & Definitions (31)

  • Definition 2.1: $k$-clause independent random SAT
  • Definition 2.2: Upper satisfiability threshold
  • Definition 2.3: Lower satisfiability threshold
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • ...and 21 more