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Couplage maximum

  • Chapter
Optimisation combinatoire

Part of the book series: Collection IRIS ((IRIS))

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Résumé

La théorie du couplage est un sujet classique et trés important de la théorie des graphes et de l’optimisation combinatoire. Tous les graphes dans ce chapitre seront non orientés. Rappelons qu’un couplage est un ensemble d’arêtes deux á deux non incidentes á un même sommet. Notre probléme est le suivant:

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Références

Littérature générale

  • 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 

  • Lovász, L., Plummer, M.D. [1986]: Matching Theory. Akadémiai Kiadó, Budapest 1986, and North-Holland, Amsterdam 1986

    MATH  Google Scholar 

  • Papadimitriou, C.H., Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, Chapter 10

    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 

  • Schrijver, A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 16 and 24

    MATH  Google Scholar 

  • Tarjan, R.E. [1983]: Data Structures and Network Algorithms. SIAM, Philadelphia 1983, Chapter 9

    Google Scholar 

Références citées

  • Alt, H., Blum, N., Mehlhorn, K., Paul, M. [1991]: Computing a maximum cardinality matching in a bipartite graph in time \( O\left( {n^{1.5} \sqrt {m/\log n} } \right) \) Information Processing Letters 37 (1991), 237–240

    Article  MATH  MathSciNet  Google Scholar 

  • Anderson, I. [1971]: Perfect matchings of a graph. Journal of Combinatorial Theory B 10 (1971), 183–186

    Article  MATH  Google Scholar 

  • Berge, C. [1957]: Two theorems in graph theory. Proceedings of the National Academy of Science of the U.S. 43 (1957), 842–844

    Article  MATH  MathSciNet  Google Scholar 

  • Berge, C. [1958]: Sur le couplage maximum d’un graphe. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris) Sér. I Math. 247 (1958), 258–259

    MATH  MathSciNet  Google Scholar 

  • Brégman, L.M. [1973]: Certain properties of nonnegative matrices and their permanents. Doklady Akademii Nauk SSSR 211 (1973), 27–30 [in Russian]. English translation: Soviet Mathematics Doklady 14 (1973), 945–949

    MathSciNet  Google Scholar 

  • Dilworth, R.P. [1950]: A decomposition theorem for partially ordered sets. Annals of Mathematics 51 (1950), 161–166

    Article  MathSciNet  Google Scholar 

  • Edmonds, J. [1965]: Paths, trees, and flowers. Canadian Journal of Mathematics 17 (1965), 449–467

    MATH  MathSciNet  Google Scholar 

  • Egoryčev, G.P. [1980]: Solution of the Van der Waerden problem for permanents. Soviet Mathematics Doklady 23 (1982), 619–622

    Google Scholar 

  • Erdős, P., Gallai, T. [1961]: On the minimal number of vertices representing the edges of a graph. Magyar Tudományos Akadémia; Matematikai Kutató Intézetének Közleményei 6 (1961), 181–203

    Google Scholar 

  • Falikman, D.I. [1981]: A proof of the Van der Waerden conjecture on the permanent of a doubly stochastic matrix. Matematicheskie Zametki 29 (1981), 931–938 [in Russian]. English translation: Math. Notes of the Acad. Sci. USSR 29 (1981), 475–479

    MathSciNet  Google Scholar 

  • Feder, T., Motwani, R. [1995]: Clique partitions, graph compression and speeding-up algorithms. Journal of Computer and System Sciences 51 (1995), 261–272

    Article  MATH  MathSciNet  Google Scholar 

  • Fremuth-Paeger, C., Jungnickel, D. [2003]: Balanced network flows VIII: a revised theory of phase-ordered algorithms and the \( O\left( {\sqrt n m \log (n^2 /m)/\log n} \right) \) bound for the nonbipartite cardinality matching problem. Networks 41 (2003), 137–142

    Article  MATH  MathSciNet  Google Scholar 

  • Frobenius, G. [1917]: Ü ber zerlegbare Determinanten. Sitzungsbericht der Königlich Preussischen Akademie der Wissenschaften XVIII (1917), 274–277

    Google Scholar 

  • Fulkerson, D.R. [1956]: Note on Dilworth’s decomposition theorem for partially ordered sets. Proceedings of the AMS 7 (1956), 701–702

    Article  MATH  MathSciNet  Google Scholar 

  • Gallai, T. [1959]: Ü ber extreme Punkt-und Kantenmengen. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae; Sectio Mathematica 2 (1959), 133–138

    MATH  MathSciNet  Google Scholar 

  • Gallai, T. [1964]: Maximale Systeme unabhängiger Kanten. Magyar Tudományos Akadémia; Matematikai Kutató Intézetének Közleményei 9 (1964), 401–413

    MATH  MathSciNet  Google Scholar 

  • Geelen, J.F. [2000]: An algebraic matching algorithm. Combinatorica 20 (2000), 61–70

    Article  MATH  MathSciNet  Google Scholar 

  • Geelen, J. Iwata, S. [2005]: Matroid matching via mixed skew-symmetric matrices. Combinatorica 25 (2005), 187–215

    Article  MATH  MathSciNet  Google Scholar 

  • Goldberg, A.V., Karzanov, A.V. [2004]: Maximum skew-symmetric flows and matchings. Mathematical Programming A 100 (2004), 537–568

    MATH  MathSciNet  Google Scholar 

  • Hall, P. [1935]: On representatives of subsets. Journal of the London Mathematical Society 10 (1935), 26–30

    Article  MATH  Google Scholar 

  • Halmos, P.R., Vaughan, H.E. [1950]: The marriage problem. American Journal of Mathematics 72 (1950), 214–215

    Article  MATH  MathSciNet  Google Scholar 

  • Hopcroft, J.E., Karp, R.M. [1973]: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing 2 (1973), 225–231

    Article  MATH  MathSciNet  Google Scholar 

  • König, D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465

    Article  MATH  MathSciNet  Google Scholar 

  • König, D. [1931]: Graphs and matrices. Matematikaiés Fizikai Lapok 38 (1931), 116–119 [in Hungarian]

    MATH  Google Scholar 

  • König, D. [1933]: Über trennende Knotenpunkte in Graphen (nebst Anwendungen auf Determinanten und Matrizen). Acta Litteratum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae (Szeged). Sectio Scientiarum Mathematicarum 6 (1933), 155–179

    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 

  • Lovász, L. [1972]: A note on factor-critical graphs. Studia Scientiarum Mathematicarum Hungarica 7 (1972), 279–280

    MathSciNet  Google Scholar 

  • Lovász, L. [1979]: On determinants, matchings and random algorithms. In: Fundamentals of Computation Theory (L. Budach, ed.), Akademie-Verlag, Berlin 1979, pp. 565–574

    Google Scholar 

  • Mendelsohn, N.S., Dulmage, A.L. [1958]: Some generalizations of the problem of distinct representatives. Canadian Journal of Mathematics 10 (1958), 230–241

    MATH  MathSciNet  Google Scholar 

  • Micali, S., Vazirani, V.V. [1980]: An O(V 1/2 E) algorithm for finding maximum matching in general graphs. Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science (1980), 17–27

    Google Scholar 

  • Mucha, M., Sankowski, P. [2004]: Maximum matchings via Gaussian elimination. Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004), 248–255

    Google Scholar 

  • Mulmuley, K., Vazirani, U.V., Vazirani, V.V. [1987]: Matching is as easy as matrix inversion. Combinatorica 7 (1987), 105–113

    Article  MATH  MathSciNet  Google Scholar 

  • Petersen, J. [1891]: Die Theorie der regulären Graphen. Acta Mathematica 15 (1891), 193–220

    Article  MathSciNet  Google Scholar 

  • Rabin, M.O., Vazirani, V.V. [1989]: Maximum matchings in general graphs through randomization. Journal of Algorithms 10 (1989), 557–567

    Article  MATH  MathSciNet  Google Scholar 

  • Rizzi, R. [1998]: König’s edge coloring theorem without augmenting paths. Journal of Graph Theory 29 (1998), 87

    Article  MATH  MathSciNet  Google Scholar 

  • Schrijver, A. [1998]: Counting 1-factors in regular bipartite graphs. Journal of Combinatorial Theory B 72 (1998), 122–135

    Article  MATH  MathSciNet  Google Scholar 

  • Sperner, E. [1928]: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27 (1928), 544–548

    Article  MATH  MathSciNet  Google Scholar 

  • Szegedy, B., Szegedy, C. [2006]: Symplectic spaces and ear-decomposition of matroids. Combinatorica 26 (2006), 353–377

    Article  MATH  MathSciNet  Google Scholar 

  • Szigeti, Z. [1996]: On a matroid defined by ear-decompositions. Combinatorica 16 (1996), 233–241

    Article  MATH  MathSciNet  Google Scholar 

  • Tutte, W.T. [1947]: The factorization of linear graphs. Journal of the London Mathematical Society 22 (1947), 107–111

    Article  MATH  MathSciNet  Google Scholar 

  • Vazirani, V.V. [1994]: A theory of alternating paths and blossoms for proving correctness of the \( O(\sqrt V E) \) general graph maximum matching algorithm. Combinatorica 14 (1994), 71–109

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Bernhard Korte & Jens Vygen

  2. Faculté de mathématiques, Université Paris-VI, 175, rue du Chevaleret, 75013, Paris

    Jean Fonlupt (Professeur émérite)

  3. Panthéon-Sorbonne Centre d’économie de la Sorbonne, Université Paris-I, 106-112, boulevard de l’Hôpital, 75013, Paris

    Alexandre Skoda

Authors
  1. Bernhard Korte
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  2. Jens Vygen
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  3. Jean Fonlupt
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  4. Alexandre Skoda
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Korte, B., Vygen, J., Fonlupt, J., Skoda, A. (2010). Couplage maximum. In: Optimisation combinatoire. Collection IRIS. Springer, Paris. https://doi.org/10.1007/978-2-287-99037-3_10

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  • DOI: https://doi.org/10.1007/978-2-287-99037-3_10

  • Publisher Name: Springer, Paris

  • Print ISBN: 978-2-287-99036-6

  • Online ISBN: 978-2-287-99037-3

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