Fast and Simple $(1+\epsilon)\Delta$-Edge-Coloring of Dense Graphs

TL;DR

This paper introduces a fast, simple randomized algorithm for approximately edge-coloring dense graphs with near-optimal colors, improving previous results and combining techniques from earlier work and dynamic coloring methods.

Contribution

The paper presents a novel, efficient randomized algorithm for $(1+ ext{epsilon}) imes ext{max degree}$ edge-coloring of dense graphs, improving upon prior algorithms in speed and simplicity.

Findings

Achieves $(1+ ext{epsilon}) imes ext{max degree}$ edge-coloring in $O(m ext{log}^3 ext{max degree}/ ext{epsilon}^2)$ time.

Works for graphs with maximum degree satisfying certain logarithmic bounds.

Algorithm is simple to implement and effective for practical dense graph coloring.

Abstract

Let $\epsilon \in (0, 1)$ and $n, \Delta \in \mathbb N$ be such that $\Delta = \Omega\left(\max\left\{\frac{\log n}{\epsilon},\, \left(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple graph $G$ of maximum degree $\Delta$, we present a randomized $O\left(m\,\log^3 \Delta\,/\,\epsilon^2\right)$-time algorithm that computes a proper $(1+\epsilon)\Delta$-edge-coloring of $G$ with high probability. This improves upon the best known results for a wide range of the parameters $\epsilon$, $n$, and $\Delta$. Our approach combines a flagging strategy from earlier work of the author with a shifting procedure employed by Duan, He, and Zhang for dynamic edge-coloring. The resulting algorithm is simple to implement and may be of practical interest.