Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Weighted Matching

  • Chapter
Combinatorial Optimization

Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

  • 1069 Accesses

Abstract

Nonbipartite weighted matching appears to be one of the “hardest” combinatorial optimization problems that can be solved in polynomial time. We shall extend Edmonds’ Cardinality Matching Algorithm to the weighted case and shall again obtain an O(n 3)-implementation. This algorithm has many applications, some of which are mentioned in the exercises and in Section 12.2.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

General Literature

  • Gerards, A.M.H. [1995]: Matching. In: Handbooks in Operations Research and Management Science; Volume 7: Network Models (M.O. Ball, T.L. Magnanti, C.L. Monma, G.L. Nemhauser, eds.), Elsevier, Amsterdam 1995, pp. 135–224

    Google Scholar 

  • Lawler, E.L. [1976]: Combinatorial Optimization; Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Chapters 5 and 6

    MATH  Google Scholar 

  • Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, Chapter 11

    MATH  Google Scholar 

  • Pulleyblank, W.R. [1995]: Matchings and extensions. In: Handbook of Combinatorics; Vol. 1 (R.L. Graham, M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam 1995

    Google Scholar 

Cited References

  • Balinski, M.L. [1972]: Establishing the matching polytope. Journal of Combinatorial Theory 13 (1972), 1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Ball, M.O., and Derigs, U. [1983]: An analysis of alternative strategies for implementing matching algorithms. Networks 13 (1983), 517–549

    Article  MathSciNet  MATH  Google Scholar 

  • Birkhoff, G. [1946]: Tres observaciones sobre el algebra lineal. Rev. Univ. Nac. Tucuman, Series A 5 (1946), 147–151

    MathSciNet  MATH  Google Scholar 

  • Cook, W., and Rohe, A. [1999]: Computing minimum-weight perfect matchings. INFORMS Journal of Computing 11 (1999), 138–148

    Article  MathSciNet  MATH  Google Scholar 

  • Cunningham, W.H., and Marsh, A.B. [1978]: A primal algorithm for optimum matching. Mathematical Programming Study 8 (1978), 50–72

    Article  MathSciNet  Google Scholar 

  • Edmonds, J. [1965]: Maximum matching and a polyhedron with (0,1) vertices. Journal of Research of the National Bureau of Standards B 69 (1965), 125–130

    Article  MathSciNet  MATH  Google Scholar 

  • Egerváry, E. [1931]: Matrixok kombinatorikus tulajdonságairol. Matematikai és Fizikai Lapok 38 (1931), 16–28 [in Hungarian]

    MATH  Google Scholar 

  • Gabow, H.N. [1973]: Implementation of algorithms for maximum matching on non-bipartite graphs. Ph.D. Thesis, Stanford University, Dept. of Computer Science, 1973

    Google Scholar 

  • Gabow, H.N. [1976]: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23 (1976), 221–234

    Article  MathSciNet  MATH  Google Scholar 

  • Gabow, H.N. [1990]: Data structures for weighted matching and nearest common ancestors with linking. Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms (1990), 434-443

    Google Scholar 

  • Grötschel, M., and Pulleyblank, W.R. [1981]: Weakly bipartite graphs and the max-cut problem. Operations Research Letters 1 (1981), 23–27

    Article  MathSciNet  MATH  Google Scholar 

  • Kuhn, H.W. [1955]: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2 (1955), 83–97

    Article  MathSciNet  Google Scholar 

  • Lipton, R.J., and Tarjan, R.E. [1979]: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36 (1979), 177–189

    Article  MathSciNet  MATH  Google Scholar 

  • Lipton, R.J., and Tarjan, R.E. [1979]: Applications of a planar separator theorem. SIAM Journal on Computing 9 (1980), 615–627

    Article  MathSciNet  Google Scholar 

  • Lovász, L. [1979]: Graph theory and integer programming. In: Discrete Optimization I; Annals of Discrete Mathematics 4 (P.L. Hammer, E.L. Johnson, B.H. Korte, eds.), North-Holland, Amsterdam 1979, pp. 141–158

    Google Scholar 

  • Mehlhorn, K., and Schäfer, G. [2000]: Implementation of O (nm log n) weighted matchings in general graphs: the power of data structures. In: Algorithm Engineering; WAE-2000; LNCS 1982 (S. Näher, D. Wagner, eds.), pp. 23-38

    Google Scholar 

  • von Neumann, J. [1953]: A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Contributions to the Theory of Games II; Ann. of Math. Stud. 28 (H.W. Kuhn, ed.), Princeton University Press, Princeton 1953, pp. 5–12

    Google Scholar 

  • Schrijver, A. [1983a]: Short proofs on the matching polyhedron. Journal of Combinatorial Theory B 34 (1983), 104–108

    Article  MathSciNet  MATH  Google Scholar 

  • Schrijver, A. [1983b]: Min-max results in combinatorial optimization. In: Mathematical Programming; The State of the Art — Bonn 1982 (A. Bachern, M. Grötschel, B. Korte, eds.), Springer, Berlin 1983, pp. 439–500

    Chapter  Google Scholar 

  • Varadarajan, K.R. [1998]: A divide-and-conquer algorithm for min-cost perfect matching in the plane. Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (1998), 320-329

    Google Scholar 

  • Weber, G.M. [1981]: Sensitivity analysis of optimal matchings. Networks 11 (1981), 41–56

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Lennéstraße 2, 53113, Bonn, Germany

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Korte, B., Vygen, J. (2002). Weighted Matching. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21711-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21711-5_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21713-9

  • Online ISBN: 978-3-662-21711-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation