Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Niv Buchbinder, Danny Segev, Yevgeny Tkach

Research output: Contribution to journalArticlepeer-review

Abstract

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in an arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 12-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 59-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 23+1/ϕ2≈0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best 11+ln2≈0.5906 bound by Epstein et al. (Inf Comput 259(1):31–40, 2018), it holds even for an easier model in which vertices along with their adjacent edges arrive online. As a result, we improve on the currently best upper bound of 0.6252 for the latter model, due to Wang and Wong (in: Proceedings of the 42nd ICALP, 2015).

Original languageEnglish
Pages (from-to)1781-1799
Number of pages19
JournalAlgorithmica
Volume81
Issue number5
DOIs
StatePublished - 15 May 2019

Bibliographical note

Funding Information:
Acknowledgements The research of Niv Buchbinder is supported by ISF Grant 1585/15 and US-Israel BSF Grant 2014414. The research of Danny Segev is supported by ISF Grant 148/16.

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Competitive analysis
  • Maximum matching
  • Online algorithms
  • Primal-dual method

ASJC Scopus subject areas

  • Computer Science (all)
  • Computer Science Applications
  • Applied Mathematics

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