Linear Programming in the Semi-streaming Model with Application to the Maximum Matching Problem

  • Kook Jin Ahn
  • Sudipto Guha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)

Abstract

In this paper, we study linear programming based approaches to the maximum matching problem in the semi-streaming model. The semi-streaming model has gained attention as a model for processing massive graphs as the importance of such graphs has increased. This is a model where edges are streamed-in in an adversarial order and we are allowed a space proportional to the number of vertices in a graph.

In recent years, there has been several new results in this semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. We present several techniques to adapt and optimize linear programming based approaches in the semi-streaming model with an application to the maximum matching problem. As a consequence, we improve (almost) all previous results on this problem, and also prove new results on interesting variants.

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References

  1. 1.
    Ahn, K.J., Guha, S.: Graph sparsification in the semi-streaming model. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 328–338. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Ahn, K.J., Guha, S.: Laminar families and metric embeddings: Non-bipartite maximum matching problem in the semi-streaming model (manuscript, 2011), http://arxiv.org/abs/1104.4058
  3. 3.
    Ahn, K.J., Guha, S.: Linear programming in the semi-streaming model with application to the maximum matching problem. To appear in the Proceedings of ICALP (2011), http://arxiv.org/abs/1104.2315
  4. 4.
    Arora, S., Hazan, E., Kale, S.: The multiplicative weights update method: a meta algorithm and applications (2005), http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf
  5. 5.
    Belkin, M., Niyogi, P.: Towards a theoretical foundation for laplacian based manifold methods. J. Comput. System Sci., 1289–1308 (2008)Google Scholar
  6. 6.
    Duan, R., Pettie, S.: Approximating maximum weight matching in near-linear time. In: FOCS, pp. 673–682 (2010)Google Scholar
  7. 7.
    Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards 69, 125–130 (1965)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Eggert, S., Kliemann, L., Srivastav, A.: Bipartite graph matchings in the semi-streaming model. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 492–503. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Epstein, L., Levin, A., Mestre, J., Segev, D.: Improved approximation guarantees for weighted matching in the semi-streaming model. In: Proc. of STACS, pp. 347–358 (2010)Google Scholar
  10. 10.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. Theor. Comput. Sci. 348(2-3), 207–216 (2005)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: Graph distances in the data-stream model. SIAM J. Comput. 38(5), 1709–1727 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fleischer, L.K.: Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discret. Math. 13(4), 505–520 (2000)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci. (JCSS) 55(1), 119–139 (1997)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Füredi, Z.: Maximum degree and fractional matchings in uniform hypergraphs. Combinatorica 1(2), 155–162 (1981)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Füredi, Z., Kahn, J., Seymour, P.D.: On the fractional matching polytope of a hypergraph. Combinatorica 13(2), 167–180 (1993)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: SODA, pp. 434–443 (1990)Google Scholar
  17. 17.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proc. FOCS, pp. 300–309 (1998)Google Scholar
  18. 18.
    Henzinger, M., Raghavan, P., Rajagopalan, S.: Computing on data streams (1998)Google Scholar
  19. 19.
    Hopcroft, J.E., Karp, R.M.: An n\(^{\mbox{5/2}}\) algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Jebara, T., Wang, J., Chang, S.-F.: Graph construction and b-matching for semi-supervised learning. In: Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009, pp. 441–448 (2009)Google Scholar
  21. 21.
    Kalantari, B., Shokoufandeh, A.: Approximation schemes for maximum cardinality matching. Technical Report LCSR-TR-248, Laboratory for Computer Science Research, Department of Computer Science. Rutgers University (1995)Google Scholar
  22. 22.
    McGregor, A.: Finding graph matchings in data streams. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 170–181. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Micali, S., Vazirani, V.V.: An \({O(\sqrt{|V|}|E|)}\) algorithm for finding maximum matching in general graphs. In: FOCS, pp. 17–27 (1980)Google Scholar
  24. 24.
    Muthukrishnan, S.: Data streams: Algorithms and applications. Foundations and Trends in Theoretical Computer Science 1(2) (2005)Google Scholar
  25. 25.
    Pettie, S., Sanders, P.: A simpler linear time 2/3-epsilon approximation for maximum weight matching. Inf. Process. Lett. 91(6), 271–276 (2004)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Plotkin, S.A., Shmoys, D.B., Tardos, É.: Fast approximation algorithms for fractional packing and covering problems. In: FOCS, pp. 495–504 (1991)Google Scholar
  27. 27.
    Preis, R.: Linear time 1/2 -approximation algorithm for maximum weighted matching in general graphs. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 259–269. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  28. 28.
    Vinkemeier, D.E.D., Hougardy, S.: A linear-time approximation algorithm for weighted matchings in graphs. ACM Transactions on Algorithms 1(1), 107–122 (2005)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Young, N.E.: Randomized rounding without solving the linear program. In: Proc. SODA, pp. 170–178 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kook Jin Ahn
    • 1
  • Sudipto Guha
    • 1
  1. 1.Department of Computer Information SciencesUniversity of PennsylvaniaUSA

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