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Counting random $k$-SAT near the satisfiability threshold

Zongchen Chen, Aditya Lonkar, Chunyang Wang, Kuan Yang, Yitong Yin

Abstract

We present efficient counting and sampling algorithms for random $k$-SAT when the clause density satisfies $α\le \frac{2^k}{\mathrm{poly}(k)}.$ In particular, the exponential term $2^k$ matches the satisfiability threshold $Θ(2^k)$ for the existence of a solution and the (conjectured) algorithmic threshold $2^k (\ln k) / k$ for efficiently finding a solution. Previously, the best-known counting and sampling algorithms required far more restricted densities $α\lesssim 2^{k/3}$ [He, Wu, Yang, SODA '23]. Notably, our result goes beyond the lower bound $d\gtrsim 2^{k/2}$ for worst-case $k$-SAT with bounded-degree $d$ [Bezáková et al, SICOMP '19], showing that for counting and sampling, the average-case random $k$-SAT model is computationally much easier than the worst-case model. At the heart of our approach is a new refined analysis of the recent novel coupling procedure by [Wang, Yin, FOCS '24], utilizing the structural properties of random constraint satisfaction problems (CSPs). Crucially, our analysis avoids reliance on the $2$-tree structure used in prior works, which cannot extend beyond the worst-case threshold $2^{k/2}$. Instead, we employ a witness tree similar to that used in the analysis of the Moser-Tardos algorithm [Moser, Tardos, JACM '10] for the Lovász Local lemma, which may be of independent interest. Our new analysis provides a universal framework for efficient counting and sampling for random atomic CSPs, including, for example, random hypergraph colorings. At the same time, it immediately implies as corollaries several structural and probabilistic properties of random CSPs that have been widely studied but rarely justified, including replica symmetry and non-reconstruction.

Counting random $k$-SAT near the satisfiability threshold

Abstract

We present efficient counting and sampling algorithms for random -SAT when the clause density satisfies In particular, the exponential term matches the satisfiability threshold for the existence of a solution and the (conjectured) algorithmic threshold for efficiently finding a solution. Previously, the best-known counting and sampling algorithms required far more restricted densities [He, Wu, Yang, SODA '23]. Notably, our result goes beyond the lower bound for worst-case -SAT with bounded-degree [Bezáková et al, SICOMP '19], showing that for counting and sampling, the average-case random -SAT model is computationally much easier than the worst-case model. At the heart of our approach is a new refined analysis of the recent novel coupling procedure by [Wang, Yin, FOCS '24], utilizing the structural properties of random constraint satisfaction problems (CSPs). Crucially, our analysis avoids reliance on the -tree structure used in prior works, which cannot extend beyond the worst-case threshold . Instead, we employ a witness tree similar to that used in the analysis of the Moser-Tardos algorithm [Moser, Tardos, JACM '10] for the Lovász Local lemma, which may be of independent interest. Our new analysis provides a universal framework for efficient counting and sampling for random atomic CSPs, including, for example, random hypergraph colorings. At the same time, it immediately implies as corollaries several structural and probabilistic properties of random CSPs that have been widely studied but rarely justified, including replica symmetry and non-reconstruction.

Paper Structure

This paper contains 28 sections, 40 theorems, 121 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Counting and sampling for random $k$-SAT
  3. Constraint-wise coupling and replica symmetry
  4. Technique Overview
  5. Organization
  6. Preliminaries
  7. Atomic CSPs and related notations
  8. Notations for (partial) assignments
  9. Notations for events and probability measures
  10. Pinned formula and pinned constraints
  11. The incidence hypergraph
  12. Lovász local lemma
  13. Structural properties of random hypergraphs
  14. Recursive coupling of random CSPs
  15. The coupling procedure
  16. ...and 13 more sections

Key Result

Theorem 1.2

There exists a universal constant $c \geq 1$ such that the following holds with high probability over the choice of the random $k$-SAT formula $\Phi=(V,C)\sim\Phi(k,n,\lfloor\alpha n \rfloor)$ where $0 < \alpha \le 2^k / k^c$. For any $\varepsilon>0$, there exist the following algorithms, both with

Figures (1)

  • Figure 1: The heuristic diagram in ding2022satisfiability depicts phase transitions in the geometry of the solution space of a random $k$-SAT instance as the density $\alpha$ increases from left to right.

Theorems & Definitions (91)

  • Definition 1.1: random $k$-SAT formulas
  • Theorem 1.2: counting and sampling random $k$-SAT solutions
  • Definition 1.3: Erdős-Rényi hypergraph
  • Theorem 1.4: counting and sampling random $k$-uniform hypergraph $q$-colorings
  • Remark 1.5: Comparison with worst-case bounded-degree CSPs
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8: replica symmetry
  • Theorem 1.9: replica symmetry of random $k$-SAT
  • Definition 1.10: non-reconstruction
  • ...and 81 more