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Maximum Matchings

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Combinatorial Optimization

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

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Abstract

Matching theory is one of the classical and most important topics in combinatorial theory and optimization. All the graphs in this chapter are undirected. Recall that a matching is a set of pairwise disjoint edges.

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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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

Cited References

  • Alt , H., Blum , N., Mehlhorn , K., and 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

    Google Scholar 

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

    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

    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

    Google Scholar 

  • Brègman , 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

    Google Scholar 

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

    Google Scholar 

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

    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., and 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

    Google Scholar 

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

    Google Scholar 

  • Frank , A. [1993]: Conservative weightings and ear-decompositions of graphs. Combinatorica 13 (1993), 65–81

    Google Scholar 

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

    Google Scholar 

  • Frobenius , G. [1917]: Über 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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Karzanov , A.V. [1973]: Tochnaya otsenka algoritma nakhozhdeniya maksimal’nogo potoka, primenennogo k zadache “o predstavitelyakh”. In: Voprosy Kibernetiki, Trudy Seminara po Kombinatornoĭ Matematike, Sovetskoe Radio, Moscow 1973, pp. 66–70 [in Russian]

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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

    Google Scholar 

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

    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 

  • Mądry , A. [2013]: Navigating central path with electrical flows: from flows to matchings, and back. Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (2013), 253–262

    Google Scholar 

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

    Google Scholar 

  • Micali , S., and 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., and 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., and Vazirani , V.V. [1987]: Matching is as easy as matrix inversion. Combinatorica 7 (1987), 105–113

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    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

    Google Scholar 

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Authors and Affiliations

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

    Bernhard Korte & Jens Vygen

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  1. Bernhard Korte
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  2. Jens Vygen
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Korte, B., Vygen, J. (2018). Maximum Matchings. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56039-6_10

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