(2δ - L)-edge-coloring is much easier than maximal matching in the distributed setting

Michael Elkin, Seth Pettie, Hsin Hao Su

Research output: Contribution to conferencePaperpeer-review

43 Scopus citations

Abstract

Graph coloring is a central problem in distributed computing. Both vertex- and edge-coloring problems have been extensively studied in this context. In this paper we show that a (2δ - l)-edge-coloring can be computed in time smaller than log n for any ∈ > 0, specifically, in eo(√log log n) rounds. This establishes a separation between the (2δ- l)-edge-coloring and Maximal Matching problems, as the latter is known to require ω(√log n) time [15]. No such separation is currently known between the (δ + l)-vertex-coloring and Maximal Independent Set problems. We devise a (1 + ∈)A-edge-coloring algorithm for an arbitrarily small constant ∈ > 0. This result applies whenever A > Ae, for some constant Ae which depends on e. The running time of this algorithm is O(log∗ δ +log n/δ1-0(1)). A much earlier logarithmic-time algorithm by Dubhashi, Grable and Panconesi [11] assumed δ ≥ (log n)1+ω(1). For A = (log n) 1+n(1) the running time of our algorithm is only O(log∗ n). This constitutes a drastic improvement of the previous logarithmic bound [11, 9]. Our results for (2δ - l)-edge-coloring also follows from our more general results concerning (1 - ∈)-locally sparse graphs. Specifically, we devise a (δ + l)-vertex coloring algorithm for (1 - ∈)-locally sparse graphs that runs in O(log∗ δ + log(l/e)) rounds for any ∈ > 0, provided that ∈δ = (log n)1+ω(1). We conclude that the (δ + 1)-vertex coloring problem for (1 - ∈)-locally sparse graphs can be solved in O(log(l/∈)) + eo√loglog n time. This imply our result about (2δ - l)-edge-coloring, because (2δ - 1)-edge-coloring reduces to (δ + 1)-vertex-coloring of the line graph of the original graph, and because line graphs are (1/2 + o(1))-locally sparse.

Original languageEnglish
Pages355-370
Number of pages16
StatePublished - 1 Jan 2015
Event26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States
Duration: 4 Jan 20156 Jan 2015

Conference

Conference26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
Country/TerritoryUnited States
CitySan Diego
Period4/01/156/01/15

Fingerprint

Dive into the research topics of '(2δ - L)-edge-coloring is much easier than maximal matching in the distributed setting'. Together they form a unique fingerprint.

Cite this