The Greedy Algorithm is not Optimal for On-Line Edge Coloring

05/14/2021
by   Amin Saberi, et al.
0

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 naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree Δ = 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 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 Δ = ω(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)· x_e, for a constant c>0.027.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/19/2019

Tight Bounds for Online Edge Coloring

Vizing's celebrated theorem asserts that any graph of maximum degree Δ a...
research
11/01/2021

Online Edge Coloring via Tree Recurrences and Correlation Decay

We give an online algorithm that with high probability computes a (e/e-1...
research
10/30/2020

Online Edge Coloring Algorithms via the Nibble Method

Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured ...
research
12/28/2019

Online Rainbow Coloring In Graphs

Rainbow coloring is a special case of edge coloring, where there must be...
research
02/14/2021

Simple vertex coloring algorithms

Given a graph G with n vertices and maximum degree Δ, it is known that G...
research
11/18/2020

Prague dimension of random graphs

The Prague dimension of graphs was introduced by Nesetril, Pultr and Rod...
research
11/08/2022

Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition

Greedy BST (or simply Greedy) is an online self-adjusting binary search ...

Please sign up or login with your details

Forgot password? Click here to reset