The Greedy Algorithm is \emph{not} Optimal for On-Line Edge Coloring

TL;DR

This paper proves that the greedy algorithm is not optimal for online edge coloring in high-degree graphs, providing a better competitive algorithm with a ratio of approximately 1.9 for adversarial arrivals.

Contribution

It introduces a new online algorithm that surpasses the greedy approach for general graphs with high degree, resolving a 30-year-old conjecture.

Findings

A new $(1.9+o(1))$-competitive online edge coloring algorithm for high-degree graphs.

An innovative online fractional bipartite matching algorithm with improved matching probabilities.

Resolution of the longstanding conjecture about the greedy algorithm's optimality in adversarial settings.

Abstract

Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of $2$ of the na\"ive greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree $\Delta = O(\log n)$, which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or for random arrival orders, an answer for fully general \emph{adversarial} arrivals remained elusive. We resolve this thirty-year-old conjecture in the affirmative, presenting a $(1.9+o(1))$-competitive online edge coloring algorithm for general graphs of degree $\Delta = \omega(\log n)$ under vertex arrivals. At the core of our results, and of possible independent interest, is a new online algorithm which rounds a fractional bipartite matching $x$ online under vertex arrivals, guaranteeing that each edge $e$ is matched with probability $(1/2+c)\cdot x_e$, for a constant $c>0.027$.