Distributed edge coloring in time polylogarithmic in Δ

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 (2Δ-1)-edge coloring can be computed in time poly(log Δ) + 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 Δ. We further show that in the CONGEST model, an (8+epsilon)Δ-edge coloring can be computed in poly(log Δ) + O(log* n) rounds. The best previous O(Δ)-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it allows to compute a 2^{O(1/epsilon)}Δ-edge coloring in time O(Δ^epsilon + log* n) for any epsilon in (0,1]