, Volume 81, Issue 5, pp 1781–1799 | Cite as

Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

  • Niv Buchbinder
  • Danny SegevEmail author
  • Yevgeny Tkach


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 \(\nicefrac {1}{2}\)-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 \(\nicefrac {5}{9}\)-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 \(\frac{2}{3+1/\phi ^2}\approx 0.5914\) on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best \(\frac{1}{1+\ln 2} \approx 0.5906\) 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).


Maximum matching Online algorithms Competitive analysis Primal-dual method 



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.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research, School of Mathematical SciencesTel Aviv universityTel AvivIsrael
  2. 2.Department of StatisticsUniversity of HaifaHaifaIsrael

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