Fast Coloring Despite Congested Relays

TL;DR

This paper introduces a highly efficient randomized algorithm for distance-2 coloring in the CONGEST model, significantly improving the round complexity for large degree graphs and addressing challenges in local reductions.

Contribution

It presents a novel $O( ext{log}^6 ext{log} n)$-round algorithm for distance-2 coloring in CONGEST, surpassing previous methods especially for large degrees.

Findings

Achieves exponential improvement over previous algorithms for large degree graphs.

Effectively handles local reductions in CONGEST for coloring problems.

Provides techniques potentially applicable to other local relation-based coloring problems.

Abstract

We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2 coloring in CONGEST with $\Delta^2+1$ colors. For $\Delta\gg\operatorname{poly}\log n$, this improves exponentially on the $O(\log\Delta+\operatorname{poly}\log\log n)$ algorithm of [Halld\'orsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.