The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovasz Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovasz Local Lemma and its generalizations, we have designed efficient resampling oracles. As applications of these techniques, we present new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.
@cite_24 show that a variant of Moser's algorithm gives an algorithmic proof in the variable model of the symmetric LLL. While this result is relatively limited when compared to the results above, their analysis is a clear example of forward-looking combinatorial analysis. Whereas Moser and Tardos use a argument to find witness trees in the algorithm's log'', analyze a structure: the tree of resampled events and their dependencies, looking forward in time. This viewpoint seems more natural and suitable for extensions.
@cite_24
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The algorithm for Lovasz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm from the witness tree. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree Δ has an acyclic proper edge coloring with at most ⌈3.74(Δ − 1)⌉ + 1 colors, whereas the previously known best bound was 4(Δ − 1). The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of this application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e. it constitutes a case of the asymmetric Lovasz Local Lemma.