New Constructive Aspects of the Lovasz Local Lemma

TL;DR

This paper explores new constructive aspects of the Lovasz Local Lemma, demonstrating how the Moser-Tardos algorithm can be applied efficiently in various complex combinatorial problems, including approximation and coloring tasks.

Contribution

It introduces novel algorithmic applications of the Moser-Tardos procedure, especially in cases with small slack and when some bad events can occur, expanding the LLL's practical utility.

Findings

Polynomial-time algorithms for complex combinatorial problems

First constant-factor approximation for the Santa Claus problem

Efficient algorithms for acyclic edge coloring and non-repetitive colorings

Abstract

The Lov\'{a}sz Local Lemma (LLL) states that the probability that none of a set of "bad" events happens is nonzero if the probability of each event is small compared to the number of bad events it depends on. A series of results have provided algorithms to efficiently construct structures whose existence is (non-constructively) guaranteed by the full asymmetric LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the Moser-Tardos procedure has sufficient randomness, leading to two classes of algorithmic applications. We first show that when an LLL application provides a small amount of slack, the running time of the Moser-Tardos algorithm is polynomial in the number of underlying independent variables (not events!), and can thus be used to give efficient constructions in cases where the underlying proof applies the LLL to super-polynomially many events (or where finding a bad event that holds is computationally hard). We demonstrate our method on applications including: the first constant-factor approximation algorithm for the Santa Claus problem, as well as efficient algorithms for acyclic edge coloring, non-repetitive graph colorings, and Ramsey-type graphs. Second, we show applications to cases where a few of the bad events can hold, leading to the first such algorithmic applications of the LLL: MAX $k$-SAT is an illustrative example of this.