A Tight Bound for Shortest Augmenting Paths on Trees

  • Bartłomiej Bosek
  • Dariusz Leniowski
  • Piotr Sankowski
  • Anna Zych-Pawlewicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree \(T=(W\uplus B, E)\) is being revealed online, i.e., in each round one vertex from \(B\) with its incident edges arrives. It was conjectured by Chaudhuri et al. [7] that the total length of all shortest augmenting paths found is \(O(n \log n)\). In this paper we prove a tight \(O(n \log n)\) upper bound for the total length of shortest augmenting paths for trees improving over \(O(n \log ^2 n)\) bound [5].

References

  1. 1.
    Bernstein, A., Holm, J., Rotenberg, E.: Online bipartite matching with amortized \(O(\log ^2 n)\) replacements. arXiv:1707.06063 (2017)
  2. 2.
    Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: ICALP, Part I, pp. 167–179 (2015)Google Scholar
  3. 3.
    Birnbaum, B.E., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008)CrossRefGoogle Scholar
  4. 4.
    Bosek, B., Leniowski, D., Sankowski, P., Zych, A.: Online bipartite matching in oine time. In: FOCS, pp. 384–393 (2014)Google Scholar
  5. 5.
    Bosek, B., Leniowski, D., Sankowski, P., Zych, A.: Shortest augmenting paths for online matchings on trees. In: WAOA, pp. 59–71 (2015)Google Scholar
  6. 6.
    Bosek, B., Leniowski, D., Sankowski, P., Zych-Pawlewicz, A.: A tight bound for shortest augmenting paths on trees. arXiv:1704.02093v2 (2017)
  7. 7.
    Chaudhuri, K., Daskalakis, C., Kleinberg, R.D., Lin, H.: Online bipartite perfect matching with augmentations. In: INFOCOM, pp. 1044–1052 (2009)Google Scholar
  8. 8.
    Devanur, N.R., Jain, K., Kleinberg, R.D.: Randomized primal-dual analysis of RANKING for online bipartite matching. In: SODA, pp. 101–107 (2013)Google Scholar
  9. 9.
    Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)CrossRefMATHGoogle Scholar
  10. 10.
    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.) WADS 1995. LNCS, vol. 955, pp. 194–205. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60220-8_62 CrossRefGoogle Scholar
  11. 11.
    Gupta, A., Kumar, A., Stein, C.: Maintaining assignments online: matching, scheduling, and flows. In: SODA, pp. 468–479 (2014)Google Scholar
  12. 12.
    Gupta, M., Peng, R.: Fully dynamic (\(1\!+\!\varepsilon \))-approximate matchings. In: FOCS, pp. 548–557 (2013)Google Scholar
  13. 13.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: STOC, pp. 352–358 (1990)Google Scholar
  14. 14.
    Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. In: STOC, pp. 745–754 (2013)Google Scholar
  15. 15.
    Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: SODA, pp. 118–126 (2007)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Bartłomiej Bosek
    • 1
  • Dariusz Leniowski
    • 2
  • Piotr Sankowski
    • 2
  • Anna Zych-Pawlewicz
    • 2
  1. 1.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Institute of Computer ScienceUniversity of WarsawWarsawPoland

Personalised recommendations