Abstract
The shortest augmenting path (\(\textsc {Sap}\)) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp in 1972 have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although is has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph \(G=(W\uplus B, E)\) is being revealed online, i.e., in each round one vertex from \(B\) with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by \(\textsc {Sap}\) is \(O(n \log n)\). However, no better bound than \(O(n^2)\) is known even for trees. In this paper we prove an \(O(n \log ^2n)\) upper bound for the total length of augmenting paths for trees.
This work was supported by NCN Grant 2013/11/D/ST6/03100, ERC StG project PAAl 259515 and FET IP project MULTIPLEX 317532.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Baswana, S., Gupta, M., Sen, S.: Fully dynamic maximal matching in \({O}(\log n)\) update time. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, ppp. 383–392. IEEE Computer Society, Washington, DC, USA (2011)
Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 167–179. Springer, Heidelberg (2015)
Bosek, B., Leniowski, D., Sankowski, P., Zych, A.: Online bipartite matching in offline time. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, 18–21 October 2014, pp. 384–393. IEEE Computer Society (2014)
Chaudhuri, K., Daskalakis, C., Kleinberg, R.D., Lin, H.: Online bipartite perfect matching with augmentations. In: INFOCOM 2009, 28th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies, 19–25 April 2009, Rio de Janeiro, Brazil, pp. 1044–1052. IEEE (2009)
Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)
Grove, E.F., Kao, M.Y., Krishnan, P., Vitter, J.S.: Online perfect matching and mobile computing. In: Akl, S.G., Dehne, F., Sack, J.-R., Santoro, N. (eds.) Algorithms and Data Structures. Lecture Notes in Computer Science, vol. 955, pp. 194–205. Springer, Heidelberg (1995)
Gupta, A., Kumar, A., Stein, C.: Maintaining assignments online: matching, scheduling, and flows. In: Chekuri, C., (ed.) Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, 5–7 January 2014, pp. 468–479. SIAM (2014)
Gupta, M., Peng, R.: Fully dynamic \((1+e)\)-approximate matchings. In: IEEE 54th Annual Symposium on Foundations of Computer. Science, pp 548–557 (2013)
Ivković, Z., Lloyd, E.L.: Fully dynamic maintenance of vertex cover. In: Leeuwen, J. (ed.) Graph-Theoretic Concepts in Computer Science. Lecture Notes in Computer Science, vol. 790, pp. 99–111. Springer, Heidelberg (1994)
Karp, R.M., Upfal, E., Wigderson, A.: Constructing a perfect matching is in random NC. Combinatorica 6(1), 35–48 (1986)
Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized on-line matching. In: 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005, pp. 264–273, October 2005
Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC 2013, pp. 745–754. ACM, New York, NY, USA (2013)
Onak, K., Rubinfeld, R.: Dynamic approximate vertex cover and maximum matching. In: Goldreich, O. (ed.) Property Testing, vol. 6390, pp. 341–345. Springer, Heidelberg (2010)
Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 118–126. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2007)
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Bosek, B., Leniowski, D., Sankowski, P., Zych, A. (2015). Shortest Augmenting Paths for Online Matchings on Trees. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-28684-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28683-9
Online ISBN: 978-3-319-28684-6
eBook Packages: Computer ScienceComputer Science (R0)