Asymptotically optimal frugal colouring

Molloy, Michael; Reed, Bruce

doi:10.1137/1.9781611973068.12

Cited by 3 publications

(3 citation statements)

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“…In particular, the last one gave an upper bound of O(log ∆/ log log ∆) on the frugality of (∆ + 1)-coloring. This is optimal up to a constant factor, because it matches lower bound of Ω(log ∆/ log log ∆) given by Hind et al [21]. However, it is not obvious whether it can be implemented efficiently in a distributed fashion, because they used a structural decomposition computed by a sequential algorithm.…”

Section: β-Frugal (∆ + 1)-coloringmentioning

confidence: 52%

“…With some modifications to their proof we show that a O(log 2 ∆/ log log ∆)-frugal (∆ + 1)-coloring can be computed in O(log n) time. Notice that the best existential bound on the frugality for (∆ + 1)-coloring is O(log ∆/ log log ∆) by Molloy and Reed [21].…”

Section: New Applicationsmentioning

confidence: 99%

“…The frugal (∆ + 1)-coloring problem for general graphs is studied by Hind et al [13], Pemmaraju and Srinivasan [27], and Molloy and Reed [21]. In particular, the last one gave an upper bound of O(log ∆/ log log ∆) on the frugality of (∆ + 1)-coloring.…”

Section: β-Frugal (∆ + 1)-coloringmentioning

confidence: 99%

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Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

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The Lovász Local Lemma (LLL), introduced by Erdős and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires O(log 2 n) rounds of communication in a distributed network.In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser-Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When epd 2 < 1 we give a truly simple LLL algorithm running in O(log 1/epd 2 n) rounds. Under the tighter condition ep(d + 1) < 1, we give a slightly slower algorithm running in O(log 2 d · log 1/ep(d+1) n) rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition p · f (d) < 1, where f (d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires Ω(log * n) rounds. In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, we give logarithmic-time *

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Section: β-Frugal (∆ + 1)-coloringmentioning

confidence: 52%

Section: New Applicationsmentioning

confidence: 99%

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Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

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Star Chromatic Index

Dvořák

Mohar

Šámal

2012

Journal of Graph Theory

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The star chromatic index χs′(G) of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi‐colored. We obtain a near‐linear upper bound in terms of the maximum degree Δ=Δ(G). Our best lower bound on χnormals′ in terms of Δ is 2Δ(1+o(1)) valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

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