Abstract
We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the Online Preemptive model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms [11, 15]. Epstein et al. [5] gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound of 1.693 for MCM. The same lower bound applies for MWM.
In this paper we show that some of the results can be improved in the case of trees and some special classes of graphs. In the online preemptive model, we present a 64/33-competitive (in expectation) randomized algorithm (which uses only two bits of randomness) for MCM on trees.
Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a “worst case” update time of O(1), in the incremental dynamic graph model, which is 3/2-approximate (in expectation) on trees, and 1.8-approximate (in expectation) on general graphs with maximum degree 3.
We also present a minor result for MWM in the online preemptive model, a 3-competitive (in expectation) randomized algorithm (that uses only O(1) bits of randomness) on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).
S. Tirodkar—A part of this work was done when the author was a student in the Department of Computer Science and Engineering at IIT Bombay.
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Acknowledgements
The first author would like to thank Ashish Chiplunkar for helpful suggestions to improve the competitive ratio of Algorithm 3, and also to improve the presentation of Sect. 4.
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Tirodkar, S., Vishwanathan, S. (2017). Maximum Matching on Trees in the Online Preemptive and the Incremental Dynamic Graph Models. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_42
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