Maximum Matching in the Online Batch-Arrival Model

  • Euiwoong Lee
  • Sahil SinglaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


Consider a two-stage matching problem, where edges of an input graph are revealed in two stages (batches) and in each stage we have to immediately and irrevocably extend our matching using the edges from that stage. The natural greedy algorithm is half competitive. Even though there is a huge literature on online matching in adversarial vertex arrival model, no positive results were previously known in adversarial edge arrival model.

For two-stage bipartite matching problem, we show that the optimal competitive ratio is exactly 2/3 in both the fractional and the randomized-integral models. Furthermore, our algorithm for fractional bipartite matching is instance optimal—achieves the best competitive ratio for any given first stage graph. We also study natural extensions of this problem to general graphs and to s stages, and present randomized-integral algorithms with competitive ratio \(\frac{1}{2} + 2^{-O(s)}\).

Our algorithms use a novel \(\mathbf {LP}\) and combine graph decomposition techniques with online primal-dual analysis.


Online algorithms Matching Primal-dual analysis Edmonds-Gallai decomposition Competitive ratio Semi-streaming 


  1. 1.
    Aggarwal, G., Goel, G., Karande, C., Mehta, A.: Online vertex-weighted bipartite matching & single-bid budgeted allocations. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1253–1264 (2011)Google Scholar
  2. 2.
    Blum, A., Sandholm, T., Zinkevich, M.: Online algorithms for market clearing. J. ACM (JACM) 53(5), 845–879 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, UK (2005)zbMATHGoogle Scholar
  4. 4.
    Buchbinder, N., Jain, K., Naor, J.S.: Online primal-dual algorithms for maximizing ad-auctions revenue. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-75520-3_24 CrossRefGoogle Scholar
  5. 5.
    Devanur, N.R., Hayes, T.P.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 71–78. ACM (2009)Google Scholar
  6. 6.
    Devanur, N.R., Jain, K., Kleinberg, R.D.: Randomized primal-dual analysis of ranking for online bipartite matching. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 101–107 (2013)Google Scholar
  7. 7.
    Epstein, L., Levin, A., Segev, D., Weimann, O.: Improved bounds for online preemptive matching. In: 30th International Symposium on Theoretical Aspects of Computer Science, pp. 389–399 (2013)Google Scholar
  8. 8.
    Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: beating 1–1/e. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 117–126. IEEE (2009)Google Scholar
  9. 9.
    Fiat, A., Algorithms, O.: The State of the Art (LNCS) (1998)Google Scholar
  10. 10.
    Goel, A., Kapralov, M., Khanna, S.: On the communication and streaming complexity of maximum bipartite matching. In: Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 468–485. SIAM (2012)Google Scholar
  11. 11.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 982–991. SIAM (2008)Google Scholar
  12. 12.
    Golovin, D., Goyal, V., Polishchuk, V., Ravi, R., Sysikaski, M.: Improved approximations for two-stage min-cut and shortest path problems under uncertainty. Math. Program. 149(1–2), 167–194 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Haeupler, B., Mirrokni, V.S., Zadimoghaddam, M.: Online stochastic weighted matching: improved approximation algorithms. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) WINE 2011. LNCS, vol. 7090, pp. 170–181. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-25510-6_15 CrossRefGoogle Scholar
  14. 14.
    Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, pp. 587–596. ACM (2011)Google Scholar
  15. 15.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, pp. 352–358 (1990)Google Scholar
  16. 16.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-\(\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Korula, N., Pál, M.: Algorithms for secretary problems on graphs and hypergraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 508–520. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02930-1_42 CrossRefGoogle Scholar
  18. 18.
    Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In: Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, pp. 597–606 (2011)Google Scholar
  19. 19.
    Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: online actions based on offline statistics. Math. Oper. Res. 37(4), 559–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mehta, A.: Online matching and ad allocation. Theor. Comput. Sci. 8(4), 265–368 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM (JACM) 54(5), 22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Swamy, C., Shmoys, D.B.: Approximation algorithms for 2-stage stochastic optimization problems. ACM SIGACT News 37(1), 33–46 (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    Vazirani, V.V.: Approximation Algorithms. Springer Science & Business Media, New York (2013)Google Scholar
  24. 24.
    Wang, Y., Wong, S.C.: Two-sided online bipartite matching and vertex cover: beating the greedy algorithm. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 1070–1081. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47672-7_87 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations