The Complexity of Distributed Edge Coloring with Small Palettes

TL;DR

This paper investigates the distributed complexity of edge coloring with small palettes, providing new lower bounds, algorithms, and techniques that improve understanding of the problem in various regimes and graph structures.

Contribution

It simplifies the round elimination technique, introduces a new randomized algorithm for small palettes, and develops a fast LLL-based algorithm for trees, advancing the theoretical understanding of distributed edge coloring.

Findings

$(2\Delta-2)$-edge coloring requires $ ext{Omega}( ext{log}_\Delta ext{log} n)$ time

A randomized algorithm achieves coloring with palette size $\Delta + ilde{O}(\sqrt{\Delta})$ in $O( ext{log}\Delta imes T_{LLL})$ time

A $(1+\epsilon)\Delta$-edge coloring algorithm for trees runs in $O( ext{log} ext{log} n)$ time

Abstract

The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree $\Delta$. 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 $(2\Delta-2)$-edge coloring requires $\Omega(\log_\Delta \log n)$ time w.h.p. and $\Omega(\log_\Delta n)$ 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 $\Delta + \tilde{O}(\sqrt{\Delta})$, which is a natural barrier for randomized approaches. The running time of the algorithm is at most $O(\log\Delta \cdot T_{LLL})$, where $T_{LLL}$ 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 $(1+\epsilon)\Delta$-edge coloring algorithm for trees running in $O(\log\log n)$ time. This algorithm arises from two new results: a deterministic $O(\log n)$-time LLL algorithm for tree-structured instances, and a randomized $O(\log\log n)$-time graph shattering method for breaking the dependency graph into independent $O(\log n)$-size LLL instances. 4. A natural approach to computing $(\Delta+1)$-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 $\Omega(\Delta\log n)$. This stands in contrast to distributed algorithms for Brooks' theorem, which exploit the existence of $O(\log_\Delta n)$-length augmenting paths.