A constructive algorithm for the Lovász Local Lemma on permutations

Harris, David G.; Srinivasan, Aravind

doi:10.1137/1.9781611973402.68

Cited by 24 publications

(40 citation statements)

References 27 publications

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“…In Section 4, we give the first sub-linear algorithm for Latin transversals: one that runs in time proportional to the square root of the input length. Latin transversals and their "partial transversal" variants are well-studied in combinatorics (see, e.g., [6,8,15,24,26,36,38]), the latter of which we encounter in item (c) next.…”

Section: Repeat While There Is Some True Bad Eventmentioning

confidence: 99%

“…We wish to select a permutation π ∈ S n with the property that no color appears twice, that is, there are no distinct x, x ′ with the property that A(x, π(x)) = A(x ′ , π(x ′ )). Such a permutation is referred to as a Latin transversal; see [6,15,24] for some of the long history behind this and related notions. One can apply the Lopsided LLL to the probability space defined by a random permutation.…”

Section: Latin Transversalsmentioning

confidence: 99%

“…In [24], a variant of the MT algorithm was presented for finding such permutations in polynomial time. The algorithm is somewhat complicated to describe, but the basic idea of this algorithm is that one can resample bad-events by performing random swaps of the relevant permutation entries.…”

Section: Latin Transversalsmentioning

confidence: 99%

“…This implies that all the results about the MT-distribution do as well. This is one of the key advantages of the proof-technique developed in [24]; later works, such as [1] and [25], have developed substantially simpler and more general proofs of the convergence of the swapping MT algorithm, but these approaches do not extend to the MT-distribution results. Note that the input size to the problem is Θ(n 2 ).…”

Section: Latin Transversalsmentioning

confidence: 99%

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Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Harris

Srinivasan

2017

ACM Trans. Algorithms

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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.

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Section: Repeat While There Is Some True Bad Eventmentioning

confidence: 99%

Section: Latin Transversalsmentioning

confidence: 99%

Section: Latin Transversalsmentioning

confidence: 99%

Section: Latin Transversalsmentioning

confidence: 99%

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Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Harris

Srinivasan

2017

ACM Trans. Algorithms

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“…Building on an earlier version of this article [19], several papers have developed generic frameworks for variations of the Moser-Tardos algorithm applied to other probability spaces. In [1], Achlioptas & Iliopoulos gave an algorithm which is based on a compression analysis for a random walk; this was improved for permutations and matchings by Kolmogorov [26].…”

Section: Comparison With Other Llll Algorithmsmentioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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New bounds for the Moser‐Tardos distribution

Harris

2020

Random Struct Algorithms

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The Lovász Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good "local" properties. In many cases, one wants more information about these structures, other than that they exist. In such case, using the "LLL-distribution", one can show that the resulting combinatorial structures have good global properties in expectation.While the LLL in its classical form is a general existential statement about probability spaces, nearly all applications in combinatorics have been turned into efficient algorithms. The simplest, variable-based setting of the LLL was covered by the seminal algorithm of Moser & Tardos (2010). This was extended by Harris & Srinivasan (2014) to random permutations, and more recently by Achlioptas & Ilioupoulos (2014) and Harvey & Vondrák (2015) to general probability spaces.One can similarly define for these algorithms an "MT-distribution," which is the distribution at the termination of the Moser-Tardos algorithm. Haeupler et al. (2011) showed bounds on the MT-distribution which essentially match the LLL-distribution for the variable-assignment setting; Harris & Srinivasan showed similar results for the permutation setting.In this work, we show new bounds on the MT-distribution which are significantly stronger than those known to hold for the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event or singleton event. As a consequence, in k-SAT instances with bounded variable occurrence, the MT-distribution satisfies an -approximate independence condition asymptotically stronger than the LLL-distribution. We use this to show a nearly tight bound on the minimum implicate size of a CNF boolean formula. Another noteworthy application is constructing independent transversals which avoid a given subset of vertices; this provides a constructive analogue to a result of Rabern (2014).In the permutation LLL setting, we show a new type of bound which is similar to the clusterexpansion LLL criterion of Bissacot et al. (2011), but is stronger and takes advantage of the extra structure in permutations. We illustrate by showing new, stronger bounds on low-weight Latin transversals and partial Latin transversals.The Lopsided Lovász Local Lemma. Although the variable-assignment LLL is by far the most common setting in combinatorics, there are other probability spaces for which a generalized form of the LLL, known as the Lopsided Lovász Local Lemma or LLLL, applies. This was introduced by Erdős & Spencer [6], which showed that a form of the LLLL applies to the probability space defined by the uniform distribution on permutations of n letters. Erdős & Spencer used this to construct Latin transversals; the space of random permutations has since been used in a variety of other combinatorial constructions. While the space of random permutations (which we refer to as the permutation LLL) is the most well-known application of the LLLL, it also covers p...

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A constructive algorithm for the Lovász Local Lemma on permutations

Cited by 24 publications

References 27 publications

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

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New bounds for the Moser‐Tardos distribution

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