License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.DISC.2017.18
URN: urn:nbn:de:0030-drops-79732
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7973/
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Fischer, Manuela ; Ghaffari, Mohsen

Sublogarithmic Distributed Algorithms for Lovász Local Lemma, and the Complexity Hierarchy

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Abstract

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of LOCAL distributed algorithms. In a recent enlightening revelation, Chang and Pettie [FOCS'17] showed that any LCL (on bounded degree graphs) that has an o(log n)-round randomized algorithm can be solved in T_(LLL)(n) rounds, which is the randomized complexity of solving (a relaxed variant of) the Lovasz Local Lemma (LLL) on bounded degree n-node graphs. Currently, the best known upper bound on T_(LLL)(n) is O(log n), by Chung, Pettie, and Su [PODC'14], while the best known lower bound is Omega(log log n), by Brandt et al. [STOC'16]. Chang and Pettie conjectured that there should be an O(log log n)-round algorithm (on bounded degree graphs).

Making the first step of progress towards this conjecture, and providing a significant improvement on the algorithm of Chung et al. [PODC'14], we prove that T_(LLL)(n)= 2^O(sqrt(log log n)). Thus, any o(log n)-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in 2^O(sqrt(log log n)) rounds.

Using this improvement and a number of other ideas, we also improve the complexity of a number of graph coloring problems (in arbitrary degree graphs) from the O(log n)-round results of Chung, Pettie and Su [PODC'14] to 2^O(sqrt(log log n)). These problems include defective coloring, frugal coloring, and list vertex-coloring.

BibTeX - Entry

@InProceedings{fischer_et_al:LIPIcs:2017:7973,
  author =	{Manuela Fischer and Mohsen Ghaffari},
  title =	{{Sublogarithmic Distributed Algorithms for Lov{\'a}sz Local Lemma, and the Complexity Hierarchy}},
  booktitle =	{31st International Symposium on Distributed Computing (DISC 2017)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-053-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{91},
  editor =	{Andr{\'e}a W. Richa},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7973},
  URN =		{urn:nbn:de:0030-drops-79732},
  doi =		{10.4230/LIPIcs.DISC.2017.18},
  annote =	{Keywords: Distributed Graph Algorithms, the Lov'{a}sz Local Lemma (LLL), Locally Checkable Labeling problems (LCL), Defective Coloring, Frugal Coloring, List Ve}
}

Keywords: Distributed Graph Algorithms, the Lov'{a}sz Local Lemma (LLL), Locally Checkable Labeling problems (LCL), Defective Coloring, Frugal Coloring, List Ve
Collection: 31st International Symposium on Distributed Computing (DISC 2017)
Issue Date: 2017
Date of publication: 12.10.2017

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