Distributed Edge Coloring in Time Polylogarithmic in $\Delta$

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 $\varepsilon\in(0,1]$.