More on online weighted edge coloring

Abstract

We revisit online weighted edge coloring. In this problem, weighted edges of a graph are presented one by one, to be colored with positive integers. It is required that for every vertex, all its edges of every common color will have a total weight not exceeding 1. We provide an improved upper bound on the performance of a greedy algorithm First Fit for the case of arbitrary weights, and for the case of weights not exceeding $\frac{1}{2}$. Here, the meaning of First Fit is that every edge is colored with a color of the smallest index that will keep the coloring valid. This improves the state-of-the-art with respect to online algorithms for this variant of edge coloring. We also show new lower bounds on the performance of any online algorithm with weights in $(0,\frac{1}{t}]$, for any integer $t\ge 2$.