A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

Brandt, Sebastian; Maus, Yannic; Uitto, Jara

doi:10.1145/3293611.3331636

Cited by 6 publications

(12 citation statements)

References 20 publications

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“…Moreover, our upper bound is tight on bounded-degree graphs due to the Ω(log * n) lower bound by Chung, Pettie and Su [CPS17]. Finally, as is the nature of upper bounds for the LLL, our result immediately implies the same upper bound for all problems that can be phrased as an LLL problem with criterion p2 d < 1, such as certain hypergraph edge-coloring problems or orientation problems in hypergraphs (see [BMU19]).…”

Section: Contributions and Techniquessupporting

confidence: 52%

“…Using this definition, the authors prove the following statement for the case of rank-3 random variables (which we give in a reformulated version using the notion of convexity instead of the concept of "incurvedness" used in [BMU19]).…”

Section: The Reductionmentioning

confidence: 99%

“…Very recently, Brandt, Maus and Uitto [BMU19] showed that if we restrict the random variables to affect at most 3 events each (which they call rank at most 3), then already under the minimally strengthened criterion p2 d < 1, there is a deterministic LLL algorithm with a complexity of O(poly d+log * n). They conjectured that this behavior also holds without their restriction on the variables.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

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Generalizing the Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

Brandt

Grunau

Rozhoň

2020

Proceedings of the 39th Symposium on Principles of Distributed Computing

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Recently, Brandt, Maus and Uitto [PODC'19] showed that, in a restricted setting, the dependency of the complexity of the distributed Lovász Local Lemma (LLL) on the chosen LLL criterion exhibits a sharp threshold phenomenon: They proved that, under the LLL criterion p2 d < 1, if each random variable affects at most 3 events, the deterministic complexity of the LLL in the LOCAL model is O(d 2 + log * n). In stark contrast, under the criterion p2 d ≤ 1, there is a randomized lower bound of Ω(log log n) by Brandt et al. [STOC'16] and a deterministic lower bound of Ω(log n) by Chang, Kopelowitz and Pettie [FOCS'16]. Brandt, Maus and Uitto conjectured that the same behavior holds for the unrestricted setting where each random variable affects arbitrarily many events.We prove their conjecture, by providing an algorithm that solves the LLL in time O(d 2 + log * n) under the LLL criterion p2 d < 1, which is tight in bounded-degree graphs due to an Ω(log * n) lower bound by Chung, Pettie and Su [PODC'14]. By the work of Brandt, Maus and Uitto, obtaining such an algorithm can be reduced to proving that all members in a certain family of functions in arbitrarily high dimensions are convex on some specific domain. Unfortunately, an analytical description of these functions is known only for dimension at most 3, which led to the aforementioned restriction of their result. While obtaining those descriptions for functions of (substantially) higher dimension seems out of the reach of current techniques, we show that their convexity can be inferred by combinatorial means.

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Section: Contributions and Techniquessupporting

confidence: 52%

Section: The Reductionmentioning

confidence: 99%

Section: Introduction 1backgroundmentioning

confidence: 99%

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Generalizing the Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

Brandt

Grunau

Rozhoň

2020

Proceedings of the 39th Symposium on Principles of Distributed Computing

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“…In fact, for the minimally more restrictive criterion < 2 −Δ , the distributed LLL can already be solved in (log * ) rounds in the LOCAL model [5,7], which implies also a probe complexity of (log * ) in the LCA model [12]. Second, we show the following general speedup theorem.…”

Section: Introductionmentioning

confidence: 76%

The Randomized Local Computation Complexity of the Lovász Local Lemma

Brandt

Grunau

Rozhoň

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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The Local Computation Algorithm (LCA) model is a popular model in the field of sublinear-time algorithms that measures the complexity of an algorithm by the number of probes the algorithm makes in the neighborhood of one node to determine that node's output.In this paper we show that the randomized LCA complexity of the Lovász Local Lemma (LLL) on constant degree graphs is Θ(log ). The lower bound follows by proving an Ω(log ) lower bound for the Sinkless Orientation problem introduced in [Brandt et al. STOC 2016]. This answers a question of [Rosenbaum, Suomela PODC 2020].Additionally, we show that every randomized LCA algorithm for a locally checkable problem with a probe complexity of ( √︁ log ) can be turned into a deterministic LCA algorithm with a probe complexity of (log * ). This improves exponentially upon the currently best known speed-up result from (log log ) to (log * ) implied by the result of [Chang, Pettie FOCS 2017] in the LOCAL model.Finally, we show that for every fixed constant ≥ 2, the deterministic VOLUME complexity of -coloring a bounded degree tree is Θ( ), where the VOLUME model is a close relative of the LCA model that was recently introduced by [Rosenbaum, Suomela PODC 2020]. CCS CONCEPTS• Theory of computation → Streaming, sublinear and near linear time algorithms.

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“…In fact, for the minimally more restrictive criterion p < 2 −∆ , the distributed LLL can already be solved in O(log * n) rounds in the LOCAL model [BMU19,BGR20], which implies also a probe complexity of O(log * n) in the LCA model [EMR14].…”

Section: Introductionmentioning

confidence: 99%

The randomized local computation complexity of the Lovász local lemma

Brandt¹,

Grunau²,

Rozhoň³

2021

Preprint

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The Local Computation Algorithm (LCA) model is a popular model in the field of sublineartime algorithms that measures the complexity of an algorithm by the number of probes the algorithm makes in the neighborhood of one node to determine that node's output.In this paper we show that the randomized LCA complexity of the Lovász Local Lemma (LLL) on constant degree graphs is Θ(log n). The lower bound follows by proving an Ω(log n) lower bound for the Sinkless Orientation problem introduced in [Brandt et al. STOC 2016]. This answers a question of [Rosenbaum, Suomela PODC 2020].Additionally, we show that every randomized LCA algorithm for a locally checkable problem with a probe complexity of o( √ log n) can be turned into a deterministic LCA algorithm with a probe complexity of O(log * n). This improves exponentially upon the currently best known speed-up result from o(log log n) to O(log * n) implied by the result of [Chang, Pettie FOCS 2017] in the LOCAL model.Finally, we show that for every fixed constant c ≥ 2, the deterministic VOLUME complexity of c-coloring a bounded degree tree is Θ(n), where the VOLUME model is a close relative of the LCA model that was recently introduced by [Rosenbaum, Suomela PODC 2020].

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A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

Cited by 6 publications

References 20 publications

Generalizing the Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

Generalizing the Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

The Randomized Local Computation Complexity of the Lovász Local Lemma

The randomized local computation complexity of the Lovász local lemma

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