An Extension of the Moser--Tardos Algorithmic Local Lemma

Pegden, Wesley

doi:10.1137/110828290

Cited by 46 publications

(70 citation statements)

References 13 publications

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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.…”

Section: The Constructive Lll For Permutationsmentioning

confidence: 86%

“…Such an event has probability 1/(n(n − 1)). We apply Pegden's criterion [32] using the weight function µ(B) = α to every bad event B, where α is a scalar to be determined.…”

Section: Latin Transversalsmentioning

confidence: 99%

“…By Lemma 3.1 and Pegden's criterion [32], a sufficient condition for the convergence of the Swapping Algorithm is that…”

Section: Latin Transversalsmentioning

confidence: 99%

“…To apply Pegden's criterion [32] for the convergence of the Swapping Algorithm, we will need to analyze the independent sets of neighbors of each bad event. To count these, it will be convenient to define the following sums.…”

Section: Rainbow Hamiltonian Cycles and Related Structuresmentioning

confidence: 99%

“…Hence an independent set of neighbors can contain one or zero of each of the four types of bad events. Using Lemma 3.1 and Pegden's criterion [32], a sufficient condition for the convergence of the Swapping Algorithm is that…”

Section: Strong Chromatic Numbermentioning

confidence: 99%

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Harris¹,

Srinivasan²

2017

Theory of Comput.

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Section: The Constructive Lll For Permutationsmentioning

confidence: 86%

“…Such an event has probability 1/(n(n − 1)). We apply Pegden's criterion [32] using the weight function µ(B) = α to every bad event B, where α is a scalar to be determined.…”

Section: Latin Transversalsmentioning

confidence: 99%

“…By Lemma 3.1 and Pegden's criterion [32], a sufficient condition for the convergence of the Swapping Algorithm is that…”

Section: Latin Transversalsmentioning

confidence: 99%

Section: Rainbow Hamiltonian Cycles and Related Structuresmentioning

confidence: 99%

Section: Strong Chromatic Numbermentioning

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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Perfect Hash Families: The Generalization to Higher Indices

Dougherty

Colbourn

2020

Springer Optimization and Its Applications

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An Extension of the Moser--Tardos Algorithmic Local Lemma

Cited by 46 publications

References 13 publications

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New bounds for the Moser‐Tardos distribution

Perfect Hash Families: The Generalization to Higher Indices

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