Online Edge Coloring of Paths and Trees with a Fixed Number of Colors

TL;DR

This paper analyzes online edge coloring of paths and trees with a fixed number of colors, proving the optimality of greedy algorithms and introducing improved randomized strategies for specific graph classes.

Contribution

It establishes the optimality of the First-Fit greedy algorithm for paths and trees, and presents a superior randomized algorithm for paths, advancing understanding of online coloring.

Findings

First-Fit is optimal for paths and trees among deterministic algorithms.

A randomized algorithm outperforms deterministic ones for paths.

Improving beyond First-Fit for trees requires unfair randomized algorithms, which are still limited.

Abstract

We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, $k$, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider $k=2$, and for trees, we consider any $k \geq 2$. We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms, on paths as well as trees. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. For trees, we prove that to obtain a better competitive ratio than First-Fit, the algorithm would have to be both randomized and unfair (i.e., reject edges that could have been colored), and even such algorithms cannot be much better than First-Fit.