Advertisement

Algorithmica

, Volume 81, Issue 5, pp 1781–1799 | Cite as

Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

  • Niv Buchbinder
  • Danny SegevEmail author
  • Yevgeny Tkach
Article
  • 46 Downloads

Abstract

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in an arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is \(\nicefrac {1}{2}\)-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a \(\nicefrac {5}{9}\)-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of \(\frac{2}{3+1/\phi ^2}\approx 0.5914\) on the competitiveness of any algorithm in the edge arrival model. Interestingly, while this result slightly falls short of the currently best \(\frac{1}{1+\ln 2} \approx 0.5906\) bound by Epstein et al. (Inf Comput 259(1):31–40, 2018), it holds even for an easier model in which vertices along with their adjacent edges arrive online. As a result, we improve on the currently best upper bound of 0.6252 for the latter model, due to Wang and Wong (in: Proceedings of the 42nd ICALP, 2015).

Keywords

Maximum matching Online algorithms Competitive analysis Primal-dual method 

Notes

Acknowledgements

The research of Niv Buchbinder is supported by ISF Grant 1585/15 and US-Israel BSF Grant 2014414. The research of Danny Segev is supported by ISF Grant 148/16.

References

  1. 1.
    Aggarwal, G., Goel, G., Karande, C., Mehta, A.: Online vertex-weighted bipartite matching and single-bid budgeted allocations. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1253–1264 (2011)Google Scholar
  2. 2.
    Azar, Y., Cohen, I.R., Roytman, A.: Online lower bounds via duality. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1038–1050 (2017)Google Scholar
  3. 3.
    Buchbinder, N., Jain, K., Naor, J.: Online primal-dual algorithms for maximizing ad-auctions revenue. In: Proceedings of the 15th Annual European Symposium on Algorithms, pp. 253–264 (2007)Google Scholar
  4. 4.
    Chiplunkar, A., Tirodkar, S., Vishwanathan, S.: On randomized algorithms for matching in the online preemptive model. In: Proceedings of the 23rd Annual European Symposium on Algorithms, pp. 325–336 (2015)Google Scholar
  5. 5.
    Devanur, N.R., Hayes, T.P.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: Proceedings 10th ACM Conference on Electronic Commerce, pp. 71–78 (2009)Google Scholar
  6. 6.
    Devanur, N.R., Jain, K.: Online matching with concave returns. In: Proceedings of the 44th ACM Symposium on Theory of Computing, pp. 137–144 (2012)Google Scholar
  7. 7.
    Devanur, N.R., Jain, K., Kleinberg, R.D.: Randomized primal-dual analysis of RANKING for online bipartite matching. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 101–107 (2013)Google Scholar
  8. 8.
    Epstein, L., Levin, A., Segev, D., Weimann, O.: Improved bounds for randomized preemptive online matching. Inf. Comput. 259(1), 31–40 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feldman, J., Mehta, A., Mirrokni, V.S., Muthukrishnan, S.: Online stochastic matching: Beating \(1-1/e\). In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 117–126 (2009)Google Scholar
  10. 10.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 982–991 (2008)Google Scholar
  11. 11.
    Guruganesh, G.P., Singla, S.: Online matroid intersection: beating half for random arrival. In: Proceedings of the 19th International Conference on Integer Programming and Combinatorial Optimization, pp. 241–253 (2017)Google Scholar
  12. 12.
    Kalyanasundaram, B., Pruhs, K.: An optimal deterministic algorithm for online b-matching. Theor. Comput. Sci. 233(1–2), 319–325 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: Proceedings of the 43rd ACM Symposium on Theory of Computing, pp. 587–596 (2011)Google Scholar
  14. 14.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pp. 352–358 (1990)Google Scholar
  15. 15.
    Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing lps. In: Proceedings of the 43rd ACM Symposium on Theory of Computing, pp. 597–606 (2011)Google Scholar
  16. 16.
    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
  17. 17.
    Mehta, A.: Online matching and ad allocation. Found. Trends Theor. Comput. Sci. 8(4), 265–368 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: Adwords and generalized online matching. J. ACM 54(5), 22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tirodkar, S., Vishwanathan, S.: Maximum matching on trees in the online preemptive and the incremental dynamic graph models. In: Proceedings of the 23rd International Conference on Computing and Combinatorics, pp. 504–515 (2017)Google Scholar
  20. 20.
    Wang, Y., Wong, S.C.: Two-sided online bipartite matching and vertex cover: beating the greedy algorithm. In: Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, pp. 1070–1081 (2015)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research, School of Mathematical SciencesTel Aviv universityTel AvivIsrael
  2. 2.Department of StatisticsUniversity of HaifaHaifaIsrael

Personalised recommendations