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

Authors Amin Saberi, David Wajc

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Author Details

Amin Saberi
  • Stanford University, CA, USA
David Wajc
  • Stanford University, CA, USA

Funding

This research is partially supported by NSF award 1812919, ONR award N000141912550, and a gift from Cisco Research.

Acknowledgements

We thank Janardhan Kulkarni for drawing our attention to [Karloff and Shmoys, 1987].

Cite AsGet BibTex

Amin Saberi and David Wajc. The Greedy Algorithm Is not Optimal for On-Line Edge Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 109:1-109:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.109

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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online algorithms
  • edge coloring
  • greedy
  • online matching

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