Improved Approximation Algorithms for Weighted Edge Coloring of Graphs

TL;DR

This paper presents improved approximation algorithms for weighted edge coloring in graphs, achieving better bounds in both online and offline settings, with specific results for certain graph classes.

Contribution

It introduces new algorithms with tighter bounds for weighted edge coloring, improving upon previous results and analyzing special graph cases.

Findings

Online algorithm uses 3.39m+o(m) colors under certain conditions.

For offline graphs with cycle structures, only m+1 colors are needed.

Multi-graph trees can be colored with 1.693m+12 colors.

Abstract

We study weighted edge coloring of graphs, where we are given an undirected edge-weighted general multi-graph $G := (V, E)$ with weights $w : E \rightarrow [0, 1]$. The goal is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. In the online setting, the edges are revealed one by one and have to be colored irrevocably as soon as they are revealed. We show that $3.39m+o(m)$ colors are enough when the maximum number of neighbors of a vertex over all the vertices is $o(m)$ and where $m$ is the maximum over all vertices of the minimum number of unit-sized bins needed to pack the weights of the incident edges to that vertex. We also prove the tightness of our analysis. This improves upon the previous best upper bound of $5m$ by Correa and Goemans [STOC 2004]. For the offline case, we show that for a simple graph with edge disjoint cycles, $m+1$ colors are sufficient and for a multi-graph tree, we show that $1.693m+12$ colors are sufficient.