Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Regularity of Random 2-SAT and 3-SAT

Andreas Basse-O'Connor, Tobias Lindhardt Overgaard, Mette Skjøtt

TL;DR

This paper quantifies exactly how under-constrained the random $k$-SAT problems are by determining their degrees of freedom, which is defined as the threshold for the number of variables the authors can fix to an arbitrary value before the problem no longer is solvable with high probability.

Abstract

We consider the random $k$-SAT problem with $n$ variables, $m=m(n)$ clauses, and clause density $α=\lim_{n\to\infty}m/n$ for $k=2,3$. It is known that if $α$ is small enough, then the random $k$-SAT problem admits a solution with high probability, which we interpret as the problem being under-constrained. In this paper, we quantify exactly how under-constrained the random $k$-SAT problems are by determining their degrees of freedom, which we define as the threshold for the number of variables we can fix to an arbitrary value before the problem no longer is solvable with high probability. We show that the random $2$-SAT and $3$-SAT problems have $n/m^{1/2}$ and $n/m^{1/3}$ degrees of freedom, respectively. Our main result is an explicit computation of the corresponding threshold functions. Our result shows that the threshold function for the random $2$-SAT problem is regular, while it is non-regular for the random $3$-SAT problem. By regular, we mean continuous and analytic on the interior of its support. This result shows that the random $3$-SAT problem is more sensitive to small changes in the clause density $α$ than the random $2$-SAT problem.

On the Regularity of Random 2-SAT and 3-SAT

TL;DR

This paper quantifies exactly how under-constrained the random -SAT problems are by determining their degrees of freedom, which is defined as the threshold for the number of variables the authors can fix to an arbitrary value before the problem no longer is solvable with high probability.

Abstract

We consider the random -SAT problem with variables, clauses, and clause density for . It is known that if is small enough, then the random -SAT problem admits a solution with high probability, which we interpret as the problem being under-constrained. In this paper, we quantify exactly how under-constrained the random -SAT problems are by determining their degrees of freedom, which we define as the threshold for the number of variables we can fix to an arbitrary value before the problem no longer is solvable with high probability. We show that the random -SAT and -SAT problems have and degrees of freedom, respectively. Our main result is an explicit computation of the corresponding threshold functions. Our result shows that the threshold function for the random -SAT problem is regular, while it is non-regular for the random -SAT problem. By regular, we mean continuous and analytic on the interior of its support. This result shows that the random -SAT problem is more sensitive to small changes in the clause density than the random -SAT problem.

Paper Structure

This paper contains 23 sections, 9 theorems, 283 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. The search tree
  5. Mixed SAT problems
  6. The threshold functions
  7. Structure
  8. Sketch of proof
  9. Preliminaries
  10. Boolean functions
  11. Literals and fixing variables
  12. Clauses and CNF formulas
  13. Fixing variables in CNF formulas
  14. Random CNF formulas
  15. The proof
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\varphi$ be a random $k$-CNF formula with $m=m(n)$ clauses and $n$ variables such that $m\to\infty$ and $m/n\to\alpha$. Let $\mathcal{L}\subseteq\pm[n]$ be a consistent set with $f=f(n)$ elements. Let finally $0\leq\gamma\leq\infty$.

Figures (3)

  • Figure 1: Finite size sampling of the threshold functions corresponding to the degrees of freedom in the random $2$- and $3$-SAT problems. Each datapoint (40 for each curve) is comprised of 2000 simulations.
  • Figure 2: Going down $f$ steps into the search tree and thus only considering assignments $x\in\{-1,1\}^n$ on one of the sub-trees branching out at $x_{f+1}$.
  • Figure 3: The threshold functions corresponding to the degrees of freedom in the random $2$-SAT and $3$-SAT problems.

Theorems & Definitions (20)

  • Definition 1: Degrees of freedom in the random $k$-SAT problem
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Corollary 3
  • proof
  • Example
  • Lemma 4
  • proof
  • ...and 10 more