Analyzing Walksat on Random Formulas

Amin Coja-Oghlan,Alan Frieze

doi:10.1137/12090191x

Abstract

Let $\mathbf{\Phi}$ be a uniformly distributed random $k$-SAT formula with $n$ variables and $m$ clauses. We prove that the \tt Walksat algorithm from Papadimitriou [On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1991, pp. 163--169] and Schöning [A probabilistic algorithm for $k$-SAT and constraint satisfaction problems, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1999, pp. 410--414] finds a satisfying assignment of $\mathbf{\Phi}$ in polynomial time with high probability if $m/n\leq\rho\cdot2^k/k$ for a certain constant $\rho>0$. This is an improvement by a factor of $\Theta(k)$ over the best previous analysis of \tt Walksat from Coja-Oghlan et al. [On smoothed $k$-CNF formulas and the Walksat algorithm, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), ACM, New York, SIAM, Philadelphia, 2009, pp. 451--460].

Highlights

Let Φ = Φk(n, m) be a k-CNF on n Boolean variables x1, . . . , xn with m clauses chosen uniformly at random (k ≥ 3)
The current rigorous results for random k-SAT mostly deal with algorithms that are extremely simple both to state and to analyze, or with algorithms that were designed so as to allow for a rigorous analysis
The present analysis techniques are essentially confined to simple algorithms that aim to construct a satisfying assignment by determining the value of one variable at a time for good, without any backtracking or reassigning variables at a later time

Summary

Introduction

2k/k, Walksat(Φ, ⌈n/k⌉) outputs a satisfying assignment w.h.p. To put Theorem 1.2 in perspective, let us compare it with other results on random k-SAT algorithms. The main point of this paper is not to produce a better algorithmic bound for random k-SAT, but to address the methodological challenge of analyzing algorithms such as Walksat that may reassign variables This difficult aspect did not occur or was sidestepped in the aforementioned previous analyses [1, 8, 9, 11]. In contrast to the previous ‘indirect’ attempts at analyzing Walksat on random formulas [4, 10], in the present paper we develop a technique for tracing the execution of the algorithm directly This allows us to keep track of the arising stochastic dependencies explicitly. Before we outline our analysis, we need some notation and preliminaries

Preliminaries

Outline of the analysis

Outline