Distributed Graph Coloring Made Easy

TL;DR

This paper introduces a simple, deterministic distributed algorithm for graph coloring that generalizes many previous results, offering scalable solutions with improved efficiency in terms of rounds needed.

Contribution

It presents a new, simple deterministic CONGEST algorithm for vertex coloring that unifies and extends several key results in distributed graph coloring.

Findings

The algorithm computes an $O(k riangle)$-vertex coloring in $O( riangle/k)+ ext{log}^* n$ rounds.

It subsumes and simplifies many existing algorithms, including Linial's color reduction.

An alternative version achieves $O(k riangle)$-coloring in $O( oot{ riangle/k})+ ext{log}^* n$ rounds.

Abstract

In this paper we present a deterministic CONGEST algorithm to compute an $O(k\Delta)$-vertex coloring in $O(\Delta/k)+\log^* n$ rounds, where $\Delta$ is the maximum degree of the network graph and $1\leq k\leq O(\Delta)$ can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it "tries colors" from the sequence in batches of size $k$. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial's color reduction [Linial, FOCS'87], the celebrated locally iterative algorithm from [Barenboim, Elkin, Goldenberg, PODC'18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between these and also simplifies the state of the art $(\Delta+1)$-coloring algorithm. At the cost of losing the full algorithm's simplicity we also provide a $O(k\Delta)$-coloring algorithm in $O(\sqrt{\Delta/k})+\log^* n$ rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for one-round color reduction algorithms.