Phase transition for Local Search on planted SAT

TL;DR

This paper analyzes the effectiveness of Local Search algorithms on planted 3-CNF formulas, revealing a phase transition at a specific clause-to-variable ratio where the algorithm's success probability sharply changes.

Contribution

It establishes a precise threshold for Local Search success on planted 3-CNF formulas based on clause-to-variable ratio, advancing understanding of local search performance in satisfiability problems.

Findings

Local Search fails whp below the threshold ratio

Local Search succeeds whp above the threshold ratio

A bound on clauses satisfied by Local Search on random 3-CNF

Abstract

The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. It is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that if there is k<7/6 such that the clause-to-variable ratio is less than k ln(n) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is k>7/6 such that the clause-to-variable ratio is greater than k ln(n)$ then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant r there is g such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio r produces an assignment that satisfies at least gn clauses less than the maximal number of satisfiable clauses.