Advertisement

Quantum Algorithms for Matching and Network Flows

  • Andris Ambainis
  • Robert Špalek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3884)

Abstract

We present quantum algorithms for some graph problems: finding a maximal bipartite matching in time \(O(n\sqrt{m}logn)\), finding a maximal non-bipartite matching in time \(O(n^2(\sqrt{m/n}+log n)log n)\), and finding a maximal flow in an integer network in time \(O(min(n^{7/6} \sqrt{m} \cdot U^{1/3},\sqrt{nU}m)log n)\), where n is the number of vertices, m is the number of edges, and Un 1/4 is an upper bound on the capacity of an edge.

Keywords

Network Flow Quantum Algorithm Classical Algorithm Graph Problem Bipartite Match 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Karzanov, A.V.: Determining the maximal flow in a network by the method of preflows. Soviet Mathematics Doklady 15, 434–437 (1974)zbMATHGoogle Scholar
  3. 3.
    Malhotra, V.M., Kumar, P., Maheshwari, S.N.: An O(V 3) algorithm for finding the maximum flows in networks. Information Processing Letters 7, 277–278 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Galil, Z., Naamad, A.: Network flow and generalized path compression. In: Proc. of 11th ACM STOC, pp. 13–26 (1979)Google Scholar
  5. 5.
    Goldberg, A.V., Rao, S.: Beyond the flow decomposition barrier. Journal of the ACM 45, 783–797 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Even, S., Tarjan, R.E.: Network flow and testing graph connectivity. SIAM Journal on Computing 4, 507–518 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Karger, D.R., Levine, M.S.: Finding maximum flows in undirected graphs seems easier than bipartite matching. In: Proc. of 30th ACM STOC, pp. 69–78 (1998)Google Scholar
  8. 8.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23, 221–234 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2, 225–231 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|} \cdot |E|)\) algorithm for finding maximum matching in general graphs. In: Proc. of 21st IEEE FOCS, pp. 17–27 (1980)Google Scholar
  12. 12.
    Mucha, M., Sankowski, P.: Maximum matchings via Gaussian elimination. In: Proc. of 45th IEEE FOCS, pp. 248–255 (2004)Google Scholar
  13. 13.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. of 28th ACM STOC, pp. 212–219 (1996)Google Scholar
  14. 14.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46, 493–505 (1998); Earlier version in Physcomp 1996CrossRefGoogle Scholar
  15. 15.
    Dürr, C., Heiligman, M., Høyer, P., Mhalla, M.: Quatum query complexity of some graph problems. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 481–493. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Berzina, A., Dubrovsky, A., Freivalds, R., Lace, L., Scegulnaja, O.: Quantum query complexity for some graph problems. In: Proc. of 30th SOFSEM, pp. 140–150 (2004)Google Scholar
  17. 17.
    Zhang, S.: On the power of Ambainis’s lower bounds. ICALP 2004 339, 241–256 (2004); Earlier version in ICALP 2004MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  19. 19.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Quantum Computation and Quantum Information: A Millennium Volume. AMS Contemporary Mathematics Series, vol. 305, pp. 53–74 (2002)Google Scholar
  20. 20.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  21. 21.
    Dinic, E.A.: Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Mathematics Doklady 11, 1277–1280 (1970)Google Scholar
  22. 22.
    Edmonds, J., Karp, R.M.: Theoretical improvement in algorithmic efficiency for network flow problems. Journal of the ACM 19, 248–264 (1972)CrossRefzbMATHGoogle Scholar
  23. 23.
    Goldberg, A.V., Rao, S.: Flows in undirected unit capacity networks. SIAM Journal on Discrete Mathematics 12, 1–5 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Andris Ambainis
    • 1
  • Robert Špalek
    • 2
  1. 1.Institute for Quantum Computing and University of WaterlooCanada
  2. 2.CWI and University of AmsterdamThe Netherlands

Personalised recommendations