Simple Local Computation Algorithms for the General Lovasz Local Lemma

TL;DR

This paper develops simple local computation algorithms for the Lovasz Local Lemma, extending the applicability of the Moser-Tardos algorithm to standard and broader LLL conditions.

Contribution

It provides the first affirmative answer that the Moser-Tardos algorithm can be adapted into local computation algorithms under the standard LLL condition.

Findings

LCAs for the Lovasz Local Lemma are achievable under standard conditions.

The techniques extend to settings beyond the standard LLL.

The algorithms are simple and efficient.

Abstract

We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully constructive, following earlier results by Beck~\cite{beck_lll} and Alon~\cite{alon_lll} giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in~\cite{Ronitt}, where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition.