We present a deterministic distributed algorithm that computes a
(2Δ−1)-edge-coloring, or even list-edge-coloring, in any n-node graph
with maximum degree Δ, in O(log7Δlogn) rounds. This answers
one of the long-standing open questions of \emph{distributed graph algorithms}
from the late 1980s, which asked for a polylogarithmic-time algorithm. See,
e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and
Elkin. The previous best round complexities were 2O(logn) by
Panconesi and Srinivasan [STOC'92] and O~(Δ)+O(log∗n)
by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our
deterministic list-edge-coloring also improves the randomized complexity of
(2Δ−1)-edge-coloring to poly(loglogn) rounds.
The key technical ingredient is a deterministic distributed algorithm for
\emph{hypergraph maximal matching}, which we believe will be of interest beyond
this result. In any hypergraph of rank r --- where each hyperedge has at most
r vertices --- with n nodes and maximum degree Δ, this algorithm
computes a maximal matching in O(r5log6+logrΔlogn) rounds.
This hypergraph matching algorithm and its extensions lead to a number of
other results. In particular, a polylogarithmic-time deterministic distributed
maximal independent set algorithm for graphs with bounded neighborhood
independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a
((logΔ/ε)O(log(1/ε)))-round deterministic
algorithm for (1+ε)-approximation of maximum matching, and a
quasi-polylogarithmic-time deterministic distributed algorithm for orienting
λ-arboricity graphs with out-degree at most (1+ε)λ,
for any constant ε>0, hence partially answering Open Problem 10 of
Barenboim and Elkin's book