New Constructive Aspects of the Lovász Local Lemma

Abstract

The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of "bad" events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is non-constructively guaranteed by the LLL, culminating in the recent breakthrough of Moser and Tardos [2010] for the full asymmetric LLL. We show that the output distribution of the Moser-Tardos algorithm well-approximates the conditional LLL-distribution, the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results.We also show that when a LLL application provides a small amount of slack, the number of resamplings of the Moser-Tardos algorithm is nearly linear 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. Even in cases where finding a bad event that holds is computationally hard, we show that applying the algorithm to avoid a polynomial-sized "core" subset of bad events leads to a desired outcome with high probability. This is shown via a simple union bound over the probabilities of noncore events in the conditional LLL-distribution, and automatically leads to simple and efficient Monte-Carlo (and in most cases RNC) algorithms. We demonstrate this idea on several applications. We give the first constant-factor approximation algorithm for the Santa Claus problem by making a LLL-based proof of Feige constructive. We provide Monte Carlo algorithms for acyclic edge coloring, nonrepetitive graph colorings, and Ramsey-type graphs. In all these applications, the algorithm falls directly out of the non-constructive LLL-based proof. Our algorithms are very simple, often provide better bounds than previous algorithms, and are in several cases the first efficient algorithms known.As a second type of application we show that the properties of the conditional LLL-distribution can be used in cases beyond the critical dependency threshold of the LLL: avoiding all bad events is impossible in these cases. As the first (even nonconstructive) result of this kind, we show that by sampling a selected smaller core from the LLL-distribution, we can avoid a fraction of bad events that is higher than the expectation. MAX k-SAT is an illustrative example of this.

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“…Some extensions of the LLL, such as the Moser-Tardos distribution bounds shown in [16], the cluster-expansion LLL criterion [9] and its connection to the Moser-Tardos algorithm [32], or the partial resampling of [18], follow almost immediately here. There are a few extensions which require slightly more discussion.…”

“…See [16] for the proof of this for the ordinary MT algorithm; the extension to the Swapping Algorithm is straightforward.…”

“…The Moser-Tardos distribution. The Witness Tree Lemma allows us to analyze the so-called "MoserTardos (MT) distribution," first discussed by [16]. The LLL and its algorithms ensure that bad events B cannot possibly occur.…”

Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions

“…In very recent work Moser and Tardos [30] and Haeupler et al [31] have shown how to make this result constructive in the sense that if P r[E i ] ≤ x i Π Ej adj. to Ei (1 − x j ) for all i they obtain a randomized algorithm that with high probability sets the base random variables such that no bad event occurs.…”

Strong LP Formulations for Scheduling Splittable Jobs on Unrelated Machines

“…This even holds for another special case called unrelated graph balancing, where each job has (possibly different) finite processing times on at most two machines [6,26]. Configuration LPs have also been studied extensively for the max-min version of the problem [2,3,8,11,17,26], which has become known as the Santa Claus problem.…”