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Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

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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 \(\nicefrac {1}{2}\)-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).

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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.

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Correspondence to Danny Segev.

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An extended abstract of this paper appeared in Proceedings of the 25th Annual European Symposium on Algorithms, pages 22:1–22:14, 2017.

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Buchbinder, N., Segev, D. & Tkach, Y. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals. Algorithmica 81, 1781–1799 (2019). https://doi.org/10.1007/s00453-018-0505-7

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