The Randomized Coloring Procedure with Symmetry-Breaking

Pemmaraju, Sriram V.; Srinivasan, Aravind

doi:10.1007/978-3-540-70575-8_26

Cited by 9 publications

(23 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Pemmaraju and Srinivasan [27] showed the existence of (∆ + 1)-colorings that are O(log 2 ∆/ log log ∆)-frugal, and proved that (log ∆ · log n/ log log n)-frugal colorings could be computed in O(log n) time. 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.…”

Section: New Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

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 *

show abstract

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%

Distributed algorithms for the Lovász local lemma and graph coloring

Chung

Pettie

2014

Proceedings of the 2014 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

show abstract

“…One paradigm that has been successfully employed in deriving several chromatic bounds is the following procedure (referred to as randomized coloring procedure in [12]) :…”

Section: Randomized Coloring Proceduresmentioning

confidence: 99%

Probabilistic Arguments in Graph Coloring (Invited Talk)

Subramanian¹

2015

Algorithms and Discrete Applied Mathematics

View full text Add to dashboard Cite

Abstract. Probabilistic arguments has come to be a powerful way to obtain bounds on various chromatic numbers. It plays an important role not only in obtaining upper bounds but also in establishing the tightness of these upper bounds. It often calls for the application of various (often simple) ideas, tools and techniques (from probability theory) like moments, concentration inequalities, known estimates on tail probabilities and various other probability estimates. A number of examples illustrate how this approach can be a very useful tool in obtaining chromatic bounds. For many of these bounds, no other approach for obtaining them is known so far. In this talk, we illustrate this approach with some specific applications to graph coloring. On the other hand, several specific applications of this approach have also motivated and led to the development of powerful tools for handling discrete probability spaces. An important tool (for establishing the tightness results) is the notion of random graphs. We do not necessarily present the best results (obtained using this approach) since the main purpose is to provide an introduction to the power and simplicity of the approach. Many of the examples and the results are already known and published in the literature and have also been improved further.

show abstract

“…The main result of that paper was to show that every graph with maximum degree ∆ in fact has a β-frugal (∆ + 1)-colouring with β = O(log 8 ∆). Pemmaraju and Srinivasan [23] recently improved this to β = O(log 2 ∆/ log log ∆).…”

Section: Introductionmentioning

confidence: 99%

“…In [29], Yuster introduced linear colorings, which are proper colourings that are both acyclic (the union of any two colour classes induces a forest) and 2-frugal; this is equivalent to saying that the union of any two colour classes is a forest of paths. In their aforementioned paper [23], Pemmaraju and Srinivasan show that every triangle-free graph has an O(log 2 ∆)-frugal O(∆/ log ∆)-colouring, and that every d-degenerate graph has a β-frugal (d + 1)-colouring for β ≈ O( ∆ d log 2 ∆). Our proof is probabilistic.…”

Section: Introductionmentioning

confidence: 99%