The complexity of distributed edge coloring depends heavily on the palette
size as a function of the maximum degree Δ. In this paper we explore the
complexity of edge coloring in the LOCAL model in different palette size
regimes.
1. We simplify the \emph{round elimination} technique of Brandt et al. and
prove that (2Δ−2)-edge coloring requires Ω(logΔlogn)
time w.h.p. and Ω(logΔn) time deterministically, even on trees.
The simplified technique is based on two ideas: the notion of an irregular
running time and some general observations that transform weak lower bounds
into stronger ones.
2. We give a randomized edge coloring algorithm that can use palette sizes as
small as Δ+O~(Δ), which is a natural barrier for
randomized approaches. The running time of the algorithm is at most
O(logΔ⋅TLLL), where TLLL is the complexity of a
permissive version of the constructive Lovasz local lemma.
3. We develop a new distributed Lovasz local lemma algorithm for
tree-structured dependency graphs, which leads to a (1+ϵ)Δ-edge
coloring algorithm for trees running in O(loglogn) time. This algorithm
arises from two new results: a deterministic O(logn)-time LLL algorithm for
tree-structured instances, and a randomized O(loglogn)-time graph
shattering method for breaking the dependency graph into independent O(logn)-size LLL instances.
4. A natural approach to computing (Δ+1)-edge colorings (Vizing's
theorem) is to extend partial colorings by iteratively re-coloring parts of the
graph. We prove that this approach may be viable, but in the worst case
requires recoloring subgraphs of diameter Ω(Δlogn). This stands
in contrast to distributed algorithms for Brooks' theorem, which exploit the
existence of O(logΔn)-length augmenting paths