Quest for Negative Dependency Graphs

Lü, Linyuan; Mohr, Austin; Székely, László Á.

doi:10.1007/978-1-4614-4565-4_21

Cited by 11 publications

(16 citation statements)

References 19 publications

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“…Other related work Applications that involve non-Cartesian spaces Ω (such as permutations, matchings and spanning trees) have often been tackled via the Lopsided LLL [6]; we refer to [14,15] for a comprehensive survey. On the level of techniques there is some connection between this paper and a recent work by Knuth [10]; we discuss this in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Commutativity in the Algorithmic Lovász Local Lemma

Kolmogorov

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We consider the recent formulation of the Algorithmic Lovász Local Lemma [9, 2] for finding objects that avoid "bad features", or "flaws". It extends the Moser-Tardos resampling algorithm [16] to more general discrete spaces. At each step the method picks a flaw present in the current state and "resamples" it using a "resampling oracle" provided by the user. However, it is less flexible than the Moser-Tardos method since [9, 2] require a specific flaw selection rule, whereas [16] allows an arbitrary rule (and thus can potentially be implemented more efficiently).We formulate a new "commutativity" condition, and prove that it is sufficient for an arbitrary rule to work. It also enables an efficient parallelization under an additional assumption. We then show that existing resampling oracles for perfect matchings and permutations do satisfy this condition.Finally, we generalize the precondition in [2] (in the case of symmetric potential causality graphs). This unifies special cases that previously were treated separately.

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Section: Introductionmentioning

confidence: 99%

Commutativity in the Algorithmic Lovász Local Lemma

Kolmogorov

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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“…In the setting of permutations, we handle the appearance of patterns in permutations as in [21]. In the settings of matchings and spanning trees, we consider the "canonical events" defined by [26], characterized by the appearance of a certain subset of edges. We also show in Section 3.5 how resampling oracles for a certain probability space can be extended in a natural way to products of such probability spaces (for example, how to go from resampling oracles for one random permutation to a collection of independent random permutations).…”

Section: Implementation Of Resampling In Specific Settingsmentioning

confidence: 99%

“…First, we appeal to a known result [26,Lemma 6] stating that given a forest F in K n with components of sizes (number of vertices) f 1 , f 2 , . .…”

Section: Lemma 35 Ifmentioning

confidence: 99%

“…However, these results are restricted to the Moser-Tardos variable model and hence cannot be viewed as algorithmic proofs of the LLL in full generality. There are many known scenarios for the LLL and its generalizations that fall outside the scope of the variable model [26,27]. Section 3 discusses several such scenarios, including random permutations, matchings and spanning trees.…”

Section: Introductionmentioning

confidence: 99%

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An Algorithmic Proof of the Lovász Local Lemma via Resampling Oracles

Harvey¹,

Vondrák²

2020

SIAM J. Comput.

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The Lovász 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 Lovász Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovász 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.

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“…Thus, in the language of the lopsided LLL, the relation ∼ defines a negative-dependence graph among the bad events. (See [27,28,30] for a study of the connection between negative dependence, random injections/permutations, and the LLLL.) Hence, the standard LLLL criterion is as follows.…”

Section: The Lopsided Lovász Local Lemma and Random Permutationsmentioning

confidence: 99%