A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching

Author Sharath Raghvendra

Thumbnail PDF


  • Filesize: 473 kB
  • 16 pages

Document Identifiers

Author Details

Sharath Raghvendra

Cite AsGet BibTex

Sharath Raghvendra. A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching. We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model. We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.
  • Online Algorithms
  • Metric Bipartite Matching


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Pankaj K. Agarwal and R. Sharathkumar. Approximation algorithms for bipartite matching with metric and geometric costs. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31-June 03, 2014, pages 555-564, 2014. Google Scholar
  2. Pankaj K. Agarwal and Kasturi R. Varadarajan. A near-linear constant-factor approximation for euclidean bipartite matching? In Proceedings of the 20th ACM Symposium on Computational Geometry, Brooklyn, New York, USA, June 8-11, 2004, pages 247-252, 2004. Google Scholar
  3. Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato. A o(n)-competitive deterministic algorithm for online matching on a line. In Approximation and Online Algorithms - 12th International Workshop, WAOA 2014, Wrocław, Poland, September 11-12, 2014, pages 11-22, 2014. Google Scholar
  4. N. Bansal, N. Buchbinder, A Madry, and J. Naor. A polylogarithmic-competitive algorithm for the k-server problem. In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 267-276, Oct 2011. Google Scholar
  5. Nikhil Bansal, Niv Buchbinder, Anupam Gupta, and Joseph Naor. An o(log² k)-competitive algorithm for metric bipartite matching. In Algorithms - ESA 2007, 15th Annual European Symposium, Eilat, Israel, October 8-10, 2007, Proceedings, pages 522-533, 2007. Google Scholar
  6. A. Gupta and K. Lewi. The online metric matching problem for doubling metrics. In Automata, Languages, and Programming, volume 7391 of LNCS, pages 424-435. Springer, 2012. Google Scholar
  7. Piotr Indyk. A near linear time constant factor approximation for euclidean bichromatic matching (cost). In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 39-42, 2007. Google Scholar
  8. Bala Kalyanasundaram and Kirk Pruhs. Online weighted matching. J. Algorithms, 14(3):478-488, 1993. Google Scholar
  9. Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC'11, pages 587-596, New York, NY, USA, 2011. ACM. Google Scholar
  10. Samir Khuller, Stephen G. Mitchell, and Vijay V. Vazirani. On-line algorithms for weighted bipartite matching and stable marriages. Theor. Comput. Sci., 127(2):255-267, 1994. Google Scholar
  11. Elias Koutsoupias and Akash Nanavati. The online matching problem on a line. In Roberto Solis-Oba and Klaus Jansen, editors, Approximation and Online Algorithms: First International Workshop, WAOA 2003, Budapest, Hungary, September 16-18, 2003. Revised Papers, pages 179-191. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004. Google Scholar
  12. Elias Koutsoupias and Christos H. Papadimitriou. On the k-server conjecture. J. ACM, 42(5):971-983, September 1995. Google Scholar
  13. M. Mahdian and Q. Yan. Online bipartite matching with random arrivals: An approach based on strongly factor-revealing lps. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC'11, pages 597-606, 2011. Google Scholar
  14. Mark S. Manasse, Lyle A. McGeoch, and Daniel D. Sleator. Competitive algorithms for server problems. J. Algorithms, 11(2):208-230, May 1990. Google Scholar
  15. A. Meyerson, A. Nanavati, and L. Poplawski. Randomized online algorithms for minimum metric bipartite matching. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, pages 954-959, 2006. Google Scholar
  16. R. Sharathkumar and Pankaj K. Agarwal. Algorithms for the transportation problem in geometric settings. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 306-317, 2012. Google Scholar
  17. R. Sharathkumar and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 385-394. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214014.