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Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Authors: Niv Buchbinder, Danny Segev, and Yevgeny Tkach

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)


Abstract
In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in 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 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 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 2/(3+1/phi^2) ~ 0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, this bound holds even for an easier model in which vertices (along with their adjacent edges) arrive online, and when the underlying graph is a tree of maximum degree at most 3.

Cite as

Niv Buchbinder, Danny Segev, and Yevgeny Tkach. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{buchbinder_et_al:LIPIcs.ESA.2017.22,
  author =	{Buchbinder, Niv and Segev, Danny and Tkach, Yevgeny},
  title =	{{Online Algorithms for Maximum Cardinality Matching with Edge Arrivals}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.22},
  URN =		{urn:nbn:de:0030-drops-78206},
  doi =		{10.4230/LIPIcs.ESA.2017.22},
  annote =	{Keywords: Maximum matching, online algorithms, competitive analysis, primal-dual method}
}
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