Deterministic Distributed (Δ + o(Δ))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity

Leonid Beranboim, Michael Elkin, Tzalik Maimon

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

14 Scopus citations

Abstract

In the distributed message-passing setting a communication network is represented by a graph whose vertices represent processors that perform local computations and communicate over the edges of the graph. In the distributed edge-coloring problem the processors are required to assign colors to edges, such that all edges incident on the same vertex are assigned distinct colors. The previouslyknown deterministic algorithms for edge-coloring employed at least (2Δ - 1) colors, even though any graph admits an edge-coloring with Δ + 1 colors [36]. Moreover, the previously-known deterministic algorithms that employed at most O(Δ) colors required superlogarithmic time [3, 6, 7, 17]. In the current paper we devise deterministic edge-coloring algorithms that employ only Δ + o(Δ) colors, for a very wide family of graphs. Specifically, as long as the arboricity a of the graph is a = O(Δ1-ϵ), for a constant ϵ > 0, our algorithm computes such a coloring within polylogarithmic deterministic time. We also devise significantly improved deterministic edge-coloring algorithms for general graphs for a very wide range of parameters. Specifically, for any value κ in the range [4Δ, 2o(log Δ) · Δ], our κ-edge-coloring algorithm has smaller running time than the best previously-known κ-edge-coloring algorithms. Our algorithms are actually much more general, since edge-coloring is equivalent to vertex-coloring of line graphs. Our method is applicable to vertexcoloring of the family of graphs with bounded diversity that contains line graphs, line graphs of hypergraphs, and many other graphs. We significantly improve upon previous vertex-coloring of such graphs, and as an implication also obtain the improved edge-coloring algorithms for general graphs. Our results are obtained using a novel technique that connects vertices or edges in a certain way that reduces clique size. The resulting structures, which we call connectors, can be colored more efficiently than the original graph. Moreover, the color classes constitute simpler subgraphs that can be colored even more efficiently using appropriate connectors. We introduce several types of connectors that are useful for various scenarios. We believe that this technique is of independent interest.

Original languageEnglish
Title of host publicationPODC 2017 - Proceedings of the ACM Symposium on Principles of Distributed Computing
PublisherAssociation for Computing Machinery
Pages175-184
Number of pages10
ISBN (Electronic)9781450349925
StatePublished - 26 Jul 2017
Event36th ACM Symposium on Principles of Distributed Computing, PODC 2017 - Washington, United States
Duration: 25 Jul 201727 Jul 2017

Conference

Conference36th ACM Symposium on Principles of Distributed Computing, PODC 2017
Country/TerritoryUnited States
CityWashington
Period25/07/1727/07/17

Keywords

  • Clique decomposition
  • Distributed coloring
  • Network partition

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