Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Abstract

Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovász Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during intermediate stages of MT. We show that these configurations have a more or less "random" form, building further on the "MT-distribution" concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving upon MT across the complexity spectrum: it makes some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial run-times in some settings, transforms certain super-polynomial-time algorithms into polynomial-time ones, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known.We show that in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution which avoids most of the bad events. We show in some cases this MT variant can run faster than the original MT algorithm itself, and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals -improving upon earlier bounds of Stein (1975) -among other applications. We furthermore give applications in enumeration, showing that most applications (where we aim for all or most of the bad events to be avoided) have large solution sets. We do this by showing that the MT-distribution has large Rényi entropy.

Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallel

“…This almost matches the best non-constructive bounds [18] which require (1 + o( 1))Δ 2 colors. In Section 7, we develop a more involved application to independent transversals.…”

“…A series of works have since improved the constant factor and provided efficient randomized algorithms. Most recently, [18] described a sequential poly-time zero-error randomized algorithm with C = Δ 2 + O(Δ 5∕3 ) colors, as well as a zero-error randomized parallel algorithm in O(log 4 n) time with a slightly larger value…”

Nonrepetitive Graph Colouring

“…Starting with the seminal work of Alon et al [8], nonrepetitive colourings of graphs have now been widely studied, including for the following graph classes: cycles [39], trees [29,61,99], outerplanar graphs [15,99], graphs with bounded treewidth [15,99], graphs with bounded degree [8,49,69,[80][81][82]131], graphs excluding a fixed immersion [148], planar graphs [14,28,47,48,86,88,88,124,125,133], graphs embeddable on a fixed surface [47,52], graphs excluding a fixed minor [47,52], graphs excluding a fixed topological minor [47,52], and graph subdivisions [17,49,69,76,106,114,122]. Table 1 summarises many of these results.…”

“…Inequality (3) was subsequently proved using a variety of techniques: the local cut lemma of Bernshteyn [19], cluster-expansion [13,24], and a novel counting argument due to Rosenfeld [131]. See [81,82] for work on efficient algorithms for finding a nonrepetitive (∆ 2 + O(∆ 5/3 ))-colouring. We present two proof of (3).…”

“…Nonrepetitive graph colouring was one of the first applications of this method that showed that entropy compression can give better results than those obtained using the Lovász Local Lemma [49,75]. • Several recent papers have presented generalised Moser-Tardos frameworks improving the original work in various ways [1,79,81,82,85]. Nonrepetitive graph colouring has been a key test case here.…”