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Abstract

We relate the problem of finding structures related to perfect matchings in bipartite graphs to a stochastic process similar to throwing balls into bins. We view each node on the left of a bipartite graph as having balls that it can throw into nodes on the right (bins) to which it is adjacent. We show that several simple algorithms based on throwing balls into bins deliver a near-perfect fractional matching, where a perfect fractional matching is a weighted subgraph on all nodes with nonnegative weights on edges so that the total weight incident at each node is 1.

Keywords

Maximum Load Bipartite Graph Perfect Match Load Vector Annual IEEE Symposium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Azar, Y., Broder, A.Z., Karlin, A.R., Upfal, E.: Balanced allocations. SIAM Journal on Computing 29, 180–200 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Azar, Y., Litichevskey, A.: Maximizing throughput in multi-queue switches. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 53–64. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Berenbrink, P., Czumaj, A., Steger, A., Vöcking, B.: Balanced allocations: The heavily loaded case. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pp. 745–754 (2000)Google Scholar
  5. 5.
    Bertsekas, D.P.: The Auction Algorithm: A Distributed Relaxation Method for the Assignment Problem. Annals of Operations Research 14, 105–123 (1988)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cherkassky, B.V., Goldberg, A.V., Martin, P., Setubal, J.C., Stolfi, J.: Augment or push: a computational study of bipartite matching and unit-capacity flow algorithms. ACM J. Exp. Algorithmics 3 (1998)Google Scholar
  7. 7.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  8. 8.
    Garg, N., Könemann, J.: Faster and Simpler Algorithms for Multicommodity Flow and other Fractional Packing Problems. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pp. 300–309 (1998)Google Scholar
  9. 9.
    Goldberg, A.V.: A natural randomization strategy for multicommodity flow and related algorithms. Information Processing Letters 42(5), 249–256 (1992)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hopcroft, J., Karp, R.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2, 225–231 (1973)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2, 83–97 (1955)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kalyanasundaram, B., Pruhs, K.R.: An optimal deterministic algorithm for online b-matching. Theoretical Computer Science 233, 319–325 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for online bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (1990)Google Scholar
  14. 14.
    Lovasz, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics. North-Holland, Amsterdam (1986)MATHGoogle Scholar
  15. 15.
    Motwani, R., Panigrahy, R., Xu, Y.: Fraction Matching via Balls-and-Bins. Technical Report (2005)Google Scholar
  16. 16.
    Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: AdWords and Generalized On-line Matching. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (2005)Google Scholar
  17. 17.
    Micali, S., Vazirani, V.: An \(O(E\sqrt V)\) algorithm for finding maximum matchings in general graphs. In: Proceedings of the 21st IEEE Symposium on the Foundations of Computer Science, pp. 17–27 (1980)Google Scholar
  18. 18.
    Panigrahy, R.: Efficient Hashing with Lookups in Two Memory Accesses. In: Proceedings of SODA (2005)Google Scholar
  19. 19.
    Plotkin, S.A., Shmoys, D., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. In: Proceedings of the 32nd Annual IEEE Symposium on the Foundations of Computer Science (1991)Google Scholar
  20. 20.
    Penn, M., Tennenholtz, M.: Constrained multi-object auctions and b-matching. Information Processing Letters 75, 29–34 (2000)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Raab, M., Steger, A.: Balls into bins – a simple and tight analysis. In: Rolim, J.D.P., Serna, M., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 159–170. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  22. 22.
    Tennenholtz, M.: Tractable combinatorial auctions and b-matching. Artif. Intell. 140, 231–243 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Rajeev Motwani
    • 1
  • Rina Panigrahy
    • 1
  • Ying Xu
    • 1
  1. 1.Dept of Computer ScienceStanford University 

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