Efficient Deterministic Distributed Coloring with Small Bandwidth

TL;DR

This paper presents the first efficient deterministic distributed algorithms for graph coloring problems in the CONGEST model, achieving polylogarithmic time complexity using network decomposition techniques.

Contribution

It introduces new deterministic algorithms for $( ext{degree}+1)$-list coloring in the CONGEST model, improving upon previous exponential-time methods and extending to the congested clique and MPC models.

Findings

Deterministic $( ext{degree}+1)$-list coloring in $O(D imes ext{polylog} n imes ext{log}^2 riangle)$ rounds.

First polylogarithmic-time deterministic algorithms for $( riangle+1)$-coloring in the CONGEST model.

Efficient algorithms for the congested clique and MPC models with logarithmic complexities.

Abstract

We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2\Delta)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $\Delta$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozho\v{n} and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(\Delta+1)$-coloring and the $(\mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(\sqrt{\log n})}$ rounds and was not based on network decompositions. Our techniques also lead to deterministic $(\mathit{degree}+1)$-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity $O(\log\Delta\cdot\log\log\Delta)$, for the MPC model, we obtain algorithms with round complexity $O(\log^2\Delta)$ for the linear-memory regime and $O(\log^2\Delta + \log n)$ for the sublinear memory regime.