New Constructive Aspects of the Lovasz Local Lemma
Bernhard Haeupler, Barna Saha, and Aravind Srinivasan

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…
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Taxonomy
TopicsComplexity and Algorithms in Graphs · Advanced Graph Theory Research · Optimization and Search Problems
