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Online Matroid Intersection: Beating Half for Random Arrival

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

Abstract

For two matroids \(\mathcal {M}_1\) and \(\mathcal {M}_2\) defined on the same ground set E, the online matroid intersection problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniformly random order. At each step, the algorithm must irrevocably decide whether to pick the element, while always maintaining a common independent set. While the natural greedy algorithm—pick an element whenever possible—is half competitive, nothing better was previously known; even for the special case of online bipartite matching in the edge arrival model. We present the first randomized online algorithm that has a \(\frac{1}{2} + \delta \) competitive ratio in expectation, where \(\delta >0\) is a constant. The expectation is over the random order and the coin tosses of the algorithm. As a corollary, we also obtain the first linear time algorithm that beats half competitiveness for offline matroid intersection.

Keywords

Online algorithms Matroid intersection Randomized algorithms Competitive analysis Linear-time algorithms 

References

  1. 1.
    Aronson, J., Dyer, M., Frieze, A., Suen, S.: Randomized greedy matching. II. Random Struct. Algorithms 6(1), 55–73 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chekuri, C., Quanrud, K.: Fast approximations for matroid intersection. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (2016)Google Scholar
  3. 3.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Structures and Their Applications, pp. 69–87 (1970)Google Scholar
  4. 4.
    Epstein, L., Levin, A., Mestre, J., Segev, D.: Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM J. Discrete Math. 25(3), 1251–1265 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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 (2008)Google Scholar
  6. 6.
    Hopcroft, J.E., Karp, R.M.: An \(n^{5/2}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huang, C.-C., Kakimura, N., Kamiyama, N.: Exact and approximation algorithms for weighted matroid intersection. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM (2016)Google Scholar
  8. 8.
    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
  9. 9.
    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
  10. 10.
    Konrad, C., Magniez, F., Mathieu, C.: Maximum matching in semi-streaming with few passes. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM -2012. LNCS, vol. 7408, pp. 231–242. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-32512-0_20 CrossRefGoogle Scholar
  11. 11.
    Korte, B., Vygen, J.: Combinatorial Optimization. Algorithms and Combinatorics, vol. 21. Springer, Berlin (2008)zbMATHGoogle Scholar
  12. 12.
    Korula, N., Mirrokni, V., Zadimoghaddam, M.: Online submodular welfare maximization: greedy beats 1/2 in random order. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, pp. 889–898 (2015)Google Scholar
  13. 13.
    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
  14. 14.
    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
  15. 15.
    Mehta, A.: Online matching and ad allocation. Theor. Comput. Sci. 8(4), 265–368 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM (JACM) 54(5), 22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mehta, A., Vazirani, V.: Personal communication (2015)Google Scholar
  18. 18.
    Oxley, J.G.: Matroid Theory, vol. 3. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  19. 19.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer Science & Business Media, Heidelberg (2002)zbMATHGoogle Scholar
  20. 20.
    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

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