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A Randomized O(log2 k)-Competitive Algorithm for Metric Bipartite Matching

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Abstract

We consider the online metric matching problem in which we are given a metric space, k of whose points are designated as servers. Over time, up to k requests arrive at an arbitrary subset of points in the metric space, and each request must be matched to a server immediately upon arrival, subject to the constraint that at most one request is matched to any particular server. Matching decisions are irrevocable and the goal is to minimize the sum of distances between the requests and their matched servers.

We give an O(log2 k)-competitive randomized algorithm for the online metric matching problem. This improves upon the best known guarantee of O(log3 k) on the competitive factor due to Meyerson, Nanavati and Poplawski (SODA ’06, pp. 954–959, 2006). It is known that for this problem no deterministic algorithm can have a competitive better than 2k−1, and that no randomized algorithm can have a competitive ratio better than lnk.

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Notes

  1. For any weighted undirected graph G=(V,E,w), it is possible to define a corresponding metric (V,d), by defining d(u,v) as the shortest path between u in v in G with edge weights w. It is easy to verify that this is indeed a metric space. A line metric is the metric induced by a line graph (with edge weights 1).

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Authors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Nikhil Bansal

  2. Computer Science Department, Open University of Israel, Raanana, Israel

    Niv Buchbinder

  3. Department of Computer Science, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Anupam Gupta

  4. Department of Computer Science, Technion, Haifa, Israel

    Joseph (Seffi) Naor

Authors
  1. Nikhil Bansal
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  2. Niv Buchbinder
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  3. Anupam Gupta
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  4. Joseph (Seffi) Naor
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Corresponding author

Correspondence to Niv Buchbinder.

Additional information

A preliminary version appeared in the 15th Annual European Symposium on Algorithms (ESA), 2007.

N. Buchbinder was supported by ISF grant 954/11 and BSF grant 2010426. A. Gupta was supported by NSF awards CCF-0964474 and CCF-1016799. J. Naor was supported by ISF grant 954/11 and BSF grant 2010426.

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Bansal, N., Buchbinder, N., Gupta, A. et al. A Randomized O(log2 k)-Competitive Algorithm for Metric Bipartite Matching. Algorithmica 68, 390–403 (2014). https://doi.org/10.1007/s00453-012-9676-9

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  • DOI: https://doi.org/10.1007/s00453-012-9676-9

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