Distributed Edge Coloring in Time Polylogarithmic in $\Delta$

TL;DR

This paper introduces new deterministic distributed algorithms for edge coloring that operate in polylogarithmic time relative to the maximum degree, improving previous results in both the LOCAL and CONGEST models.

Contribution

The authors present the first polylogarithmic-in-elta time algorithms for $(2elta-1)$-edge coloring in the LOCAL model and for $(8+psilon)elta$-edge coloring in the CONGEST model, advancing distributed symmetry breaking.

Findings

Edge coloring can be computed in polylogarithmic time in elta in the LOCAL model.

In the CONGEST model, elta-edge coloring is achievable in polylogarithmic rounds.

Improves upon previous algorithms with quasi-polylogarithmic or polynomial dependencies on elta.

Abstract

We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a $(2\Delta-1)$-edge coloring can be computed in time $\mathrm{poly}\log\Delta + O(\log^* n)$ in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on $\Delta$. We further show that in the CONGEST model, an $(8+\varepsilon)\Delta$-edge coloring can be computed in $\mathrm{poly}\log\Delta + O(\log^* n)$ rounds. The best previous $O(\Delta)$-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a $2^{O(1/\varepsilon)}\Delta$-edge coloring in time $O(\Delta^\varepsilon + \log^* n)$ for any $\varepsilon\in(0,1]$.