Optimal Matching Forests and Valuated Delta-Matroids

  • Kenjiro Takazawa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6655)


The matching forest problem in mixed graphs is a common generalization of the matching problem in undirected graphs and the branching problem in directed graphs. Giles presented an \(\mathrm{O}( n\sp{2}m )\)-time algorithm for finding a maximum-weight matching forest, where n is the number of vertices and m is that of edges, and a linear system describing the matching forest polytope. Later, Schrijver proved total dual integrality of the linear system. In the present paper, we reveal another nice property of matching forests: the degree sequences of the matching forests in any mixed graph form a delta-matroid and the weighted matching forests induce a valuated delta-matroid. We remark that the delta-matroid is not necessarily even, and the valuated delta-matroid induced by weighted matching forests slightly generalizes the well-known notion of Dress and Wenzel’s valuated delta-matroids. By focusing on the delta-matroid structure and reviewing Giles’ algorithm, we design a simpler \(\mathrm{O}( n\sp{2}m )\)-time algorithm for the weighted matching forest problem. We also present a faster \(\mathrm{O}( n\sp{3} )\)-time algorithm by using Gabow’s method for the weighted matching problem.


Matching Branching Matching Forest Delta-Matroid Valuated Delta-Matroid Primal-Dual Algorithm 


  1. 1.
    Bouchet, A.: Greedy Algorithm and Symmetric Matroids. Math. Programming 38, 147–159 (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bouchet, A.: Matchings and Δ-Matroids. Discrete Appl. Math. 24, 55–62 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Discrete Math. 71, 205–217 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cunningham, W.H., Marsh III, A.B.: A Primal Algorithm for Optimum Matching. Math. Programming Study 8, 50–72 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dress, A.W.M., Havel, T.: Some Combinatorial Properties of Discriminants in Metric Vector Spaces. Adv. Math. 62, 285–312 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dress, A.W.M., Wenzel, W.: A Greedy-Algorithm Characterization of Valuated Δ-matroids. Appl. Math. Lett. 4, 55–58 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Edmonds, J.: Maximum Matching and a Polyhedron with 0,1-Vertices. J. Res. Natl. Bur. Stand. Sect. B 69, 125–130 (1965)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Edmonds, J.: Paths, Trees, and Flowers. Canad. J. Math. 17, 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Edmonds, J.: Optimum Branchings. J. Res. Natl. Bur. Stand. Sect. B 71, 233–240 (1967)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Edmonds, J., Giles, R.: A Min-Max Relation for Submodular Functions on Graphs. Ann. Discrete Math. 1, 185–204 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Frank, A.: Covering Branchings. Acta Sci. Math (Szeged) 41, 77–81 (1979)MathSciNetMATHGoogle Scholar
  12. 12.
    Frank, A., Tardos, É.: Generalized Polymatroids and Submodular Flows. Math. Programming 42, 489–563 (1988)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gabow, H.N.: Implementation of Algorithms for Maximum Matching on Nonbipartite Graphs, Ph.D. thesis, Stanford University (1973)Google Scholar
  14. 14.
    Giles, R.: Optimum Matching Forests I: Special Weights. Math. Programming 22, 1–11 (1982)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giles, R.: Optimum Matching Forests II: General Weights. Math. Programming 22, 12–38 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Giles, R.: Optimum Matching Forests III: Facets of Matching Forest Polyhedra. Math. Programming 22, 39–51 (1982)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Keijsper, J.: A Vizing-Type Theorem for Matching Forests. Discrete Math. 260, 211–216 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lovász, L.: Matroid Matching and Some Applications. J. Combin. Theory Ser. B 28, 208–236 (1980)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Murota, K.: Characterizing a Valuated Delta-Matroid as a Family of Delta-Matroids. J. Oper. Res. Soc. Japan 40, 565–578 (1997)MathSciNetMATHGoogle Scholar
  20. 20.
    Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)CrossRefMATHGoogle Scholar
  21. 21.
    Schrijver, A.: Total Dual Integrality of Matching Forest Constraint. Combinatorica 20, 575–588 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Heidelberg (2003)MATHGoogle Scholar
  23. 23.
    Vizing, V.G.: Ob Otsenke Khromaticheskogo Klassa p-grapha (in Russian). Diskretnyĭ 3, 25–30 (1964)Google Scholar
  24. 24.
    Vizing, V.G.: Khromaticheskiĭ Klass Mul’tigrafa (in Russian). Kibernetika 1(3), 29–39 (1965)Google Scholar
  25. 25.
    Wenzel, W.: Δ-Matroids with the Strong Exchange Conditions. Appl. Math. Lett. 6, 67–70 (1993)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wenzel, W.: Pfaffian Forms and Δ-Matroids. Discrete Math. 115, 253–266 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kenjiro Takazawa
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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