A Local Lemma for Focused Stochastic Algorithms

TL;DR

This paper introduces a flexible framework for analyzing focused stochastic local search algorithms, extending Lovász Local Lemma-based methods by allowing measure distortion to evaluate progress.

Contribution

It develops a general analysis framework for focused stochastic algorithms that accommodates measure distortion, broadening applicability beyond LLL-compatible algorithms.

Findings

Framework applies to arbitrary algorithms and measures

Recovers LLL-based algorithms as a special case

Enables analysis of more general local search algorithms

Abstract

We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These are algorithms that search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that addresses the violation, while hopefully not introducing many new ones. An important class of focused local search algorithms with provable performance guarantees has recently arisen from algorithmizations of the Lov\'{a}sz Local Lemma (LLL), a non-constructive tool for proving the existence of satisfying states by introducing a background measure on the state space. While powerful, the state transitions of algorithms in this class must be, in a precise sense, perfectly compatible with the background measure. In many applications this is a very restrictive requirement and one needs to step outside the class. Here we introduce the notion of \emph{measure distortion} and develop a framework for analyzing arbitrary focused stochastic local search algorithms, recovering LLL algorithmizations as the special case of no distortion. Our framework takes as input an arbitrary such algorithm and an arbitrary probability measure and shows how to use the measure as a yardstick of algorithmic progress, even for algorithms designed independently of the measure.