Distributed Delta-Coloring under Bandwidth Limitations

TL;DR

This paper presents a near-optimal distributed algorithm for graph coloring with maximum degree Δ using limited bandwidth, achieving polyloglog n rounds, close to theoretical lower bounds.

Contribution

It introduces a novel randomized algorithm in the CONGEST model that reduces the coloring problem to Lovász local lemma instances and list coloring, improving efficiency under bandwidth constraints.

Findings

Achieves polyloglog n round complexity in the CONGEST model.

Close to the theoretical lower bound of Ω(log log n) rounds.

Uses reduction to Lovász local lemma and list coloring problems.

Abstract

We consider the problem of coloring graphs of maximum degree $\Delta$ with $\Delta$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{poly}\log\log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $\Omega(\log \log n)$ rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lov\'asz local lemma (LLL) and the $deg+1$-list coloring problem.