A Simple Algorithm for Near-Vizing Edge-Coloring in Near-Linear Time

TL;DR

This paper introduces a simple, near-linear time algorithm for approximate edge-coloring of graphs with high maximum degree, offering a practical alternative to more complex existing solutions.

Contribution

The paper presents a simpler algorithm for near-approximate edge-coloring that is easier to analyze and implement, with competitive running time for graphs with large maximum degree.

Findings

Achieves $(1+ ext{epsilon}) imes ext{Delta}$-edge-coloring in near-linear time

Simpler approach compared to previous algorithms

Potential for practical applications due to simplicity

Abstract

We present a simple $(1+\varepsilon)\Delta$-edge-coloring algorithm for graphs of maximum degree $\Delta = \Omega(\log n / \varepsilon)$ with running time $O\left(m\,\log^3 n/\varepsilon^3\right)$. Our algorithm improves upon that of [Duan, He, and Zhang; SODA19], which was the first near-linear time algorithm for this problem. While our results are weaker than the current state-of-the-art, our approach is significantly simpler, both in terms of analysis as well as implementation, and may be of practical interest.