Maximum skew-symmetric flows

  • Andrew V. Goldberg
  • Alexander V. Karzanov
Session 3. Chair: Giuseppe Italiano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 979)

Abstract

We introduce the maximum skew-symmetric flow problem which generalizes flow and matching problems. We develop a theory of skew-symmetric flows that is parallel to the classical flow theory. We use the newly developed theory to extend, in a natural way, the blocking flow method of Dinitz to the skew-symmetric flow case. In the special case of the skew-symmetric flow problem that corresponds to cardinality matching, our algorithm is simpler and more efficient than the corresponding matching algorithm.

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References

  1. 1.
    G. M. Adel'son-Vel'ski, E. A. Dinits, and A. V. Karzanov. Flow Algorithms Nauka, Moscow, 1975. In Russian.Google Scholar
  2. 2.
    R. K. Ahuja, J. B. Orlin, and R. E. Tarjan. Improved Time Bounds for the Maximum Flow Problem. SIAM J. Comput., 18:939–954, 1989.CrossRefGoogle Scholar
  3. 3.
    N. Blum. A New Approach to Maximum Matching in General Graphs. In Proc. ICALP, pages 586–597, 1990.Google Scholar
  4. 4.
    N. Blum. A New Approach to Maximum Matching in General Graphs. Technical report, Institut für Informatik der Universität Bonn, 1990.Google Scholar
  5. 5.
    J. Cheriyan, T. Hagerup, and K. Mehlhorn. Can a Maximum Flow be Computed in o(nm) Time? In Proc. ICALP, 1990.Google Scholar
  6. 6.
    D. D. Sleator and R. E. Tarjan. Self-adjusting binary search trees. J. Assoc. Comput. Mach., 32:652–686, 1985.Google Scholar
  7. 7.
    E. A. Dinic. Algorithm for Solution of a Problem of Maximum Flow in Networks with Power Estimation. Soviet Math. Dokl., 11:1277–1280, 1970.Google Scholar
  8. 8.
    J. Edmonds. Paths, Trees and Flowers. Canada J. Math., 17:449–467, 1965.Google Scholar
  9. 9.
    J. Edmonds and E. L. Johnson. Matching, a Well-Solved Class of Integer Linear Programs. In R. Guy, H. Haneni, and J. Schönhein, editors, Combinatorial Structures and Their Applications, pages 89–92. Gordon and Breach, NY, 1970.Google Scholar
  10. 10.
    J. Edmonds and R. M. Karp. Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems. J. Assoc. Comput. Mach., 19:248–264, 1972.Google Scholar
  11. 11.
    S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM J. Comput., 4:507–518, 1975.CrossRefGoogle Scholar
  12. 12.
    T. Feder and R. Motwani. Clique Partitions, Graph Compression and Speeding-up Algorithms. In Proc. 23st Annual ACM Symposium on Theory of Computing, pages 123–133, 1991.Google Scholar
  13. 13.
    L. R. Ford, Jr. and D. R. Fulkerson. Flows in Networks. Princeton Univ. Press, Princeton, NJ, 1962.Google Scholar
  14. 14.
    H. N. Gabow and R. E. Tarjan. Faster scaling algorithms for general graphmatching problems. J. Assoc. Comput. Mach., 38:815–853, 1991.Google Scholar
  15. 15.
    A. V. Goldberg and A. V. Karzanov. Path Problems in Skew-Symmetric Graphs. Technical Report STAN-CS-93-1489, Department of Computer Science, Stanford University, 1993.Google Scholar
  16. 16.
    A. V. Goldberg and A. V. Karzanov. Path Problems in Skew-Symmetric Graphs. In Proc. 5th ACM-SIAM Symposium on Discrete Algorithms, pages 526–535, 1994.Google Scholar
  17. 17.
    A. V. Goldberg and R. E. Tarjan. A New Approach to the Maximum Flow Problem. J. Assoc. Comput. Mach., 35:921–940, 1988.Google Scholar
  18. 18.
    A. V. Goldberg and R. E. Tarjan. Finding Minimum-Cost Circulations by Successive Approximation. Math. of Oper. Res., 15:430–466, 1990.Google Scholar
  19. 19.
    A. V. Karzanov. O nakhozhdenii maksimal'nogo potoka v setyakh spetsial'nogo vida i nekotorykh prilozheniyakh. In Matematicheskie Voprosy Upravleniya Proizvodstvom, volume 5. Moscow State University Press, Moscow, 1973. In Russian; title translation: On Finding Maximum Flows in Network with Special Structure and Some Applications.Google Scholar
  20. 20.
    A. V. Karzanov. Tochnaya otzenka algoritma nakhojdeniya maksimalnogo potoka, primenennogo k zadache “o predstavitelyakh”. In Problems in Cibernetics, volume 5, pages 66–70. Nauka, Moscow, 1973. In Russian; title translation: The exact time bound for a maximum flow algorithm applied to the set representatives problem.Google Scholar
  21. 21.
    V. King, S. Rao, and R. Tarjan. A Faster Deterministic Maximum Flow Algorithm. J. Algorithms, 17:447–474, 1994.CrossRefGoogle Scholar
  22. 22.
    E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Reinhart, and Winston, New York, NY., 1976.Google Scholar
  23. 23.
    L. Lovász and M. D. Plummer. Matching Theory. Akadémiai Kiadó, Budapest, 1986.Google Scholar
  24. 24.
    S. Micali and V. V. Vazirani. An O(√¦V∥E¦) algorithm for finding maximum matching in general graphs. In Proc. 21th IEEE Annual Symposium on Foundations of Computer Science, pages 17–27, 1980.Google Scholar
  25. 25.
    V. V. Vazirani. A Theory of Alternating Paths and Blossoms for Proving Correctness of the O(√VE) General Graph Maximum Matching Algorithm. Combinatorica, to appear.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Andrew V. Goldberg
    • 1
  • Alexander V. Karzanov
    • 2
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.Inst. for Systems AnalysisMoscowRussia

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