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. Our main problem is:
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
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
Lawler, E.L. [1976]: Combinatorial Optimization; Networks and Matroids. Holt, Rinehart and Winston, New York 1976, Chapters 5 and 6
Lovász, L., and Plummer, M.D. [1986]: Matching Theory. Akadémiai Kiadó, Budapest 1986, and North-Holland, Amsterdam 1986
Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, Chapter 10
Pulleyblank, W.R. [1995]: Matchings and extensions. In: Handbook of Combinatorics; Vol. 1 (R.L. Graham, M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam 1995
Schrijver, A. [2003]: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003, Chapters 16 and 24
Tarjan, R.E. [1983]: Data Structures and Network Algorithms. SIAM, Philadelphia 1983, Chapter 9
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
Anderson, I. [1971]: Perfect matchings of a graph. Journal of Combinatorial Theory B 10 (1971), 183–186
Berge, C. [1957]: Two theorems in graph theory. Proceedings of the National Academy of Science of the U.S. 43 (1957), 842–844
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
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
Dilworth, R.P. [1950]: A decomposition theorem for partially ordered sets. Annals of Mathematics 51 (1950), 161–166
Edmonds, J. [1965]: Paths, trees, and flowers. Canadian Journal of Mathematics 17 (1965), 449–467
Egoryčev, G.P. [1980]: Solution of the van der Waerden problem for permanents. Soviet Mathematics Doklady 23 (1982), 619–622
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
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
Feder, T., and Motwani, R. [1995]: Clique partitions, graph compression and speeding-up algorithms. Journal of Computer and System Sciences 51 (1995), 261–272
Fremuth-Paeger, C., and Jungnickel, D. [2003]: Balanced network flows VIII: a revised theory of phase-ordered algorithms and the \( O\left( {\sqrt n m {\text{log}}(n^2 {\text{/}}m){\text{/log }}n} \right) \) bound for the nonbipartite cardinality matching problem. Networks 41 (2003), 137–142
Frobenius, G. [1917]: Über zerlegbare Determinanten. Sitzungsbericht der Königlich Preussischen Akademie der Wissenschaften XVIII (1917), 274–277
Fulkerson, D.R. [1956]: Note on Dilworth’s decomposition theorem for partially ordered sets. Proceedings of the AMS 7 (1956), 701–702
Gallai, T. [1959]: Über extreme Punkt-und Kantenmengen. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae; Sectio Mathematica 2 (1959), 133–138
Gallai, T. [1964]: Maximale Systeme unabhängiger Kanten. Magyar Tudományos Akadémia; Matematikai Kutató Intézetének Közleményei 9 (1964), 401–413
Geelen, J.F. [2000]: An algebraic matching algorithm. Combinatorica 20 (2000), 61–70
Geelen, J. and Iwata, S. [2005]: Matroid matching via mixed skew-symmetric matrices. Combinatorica 25 (2005), 187–215
Goldberg, A.V., and Karzanov, A.V. [2004]: Maximum skew-symmetric flows and matchings. Mathematical Programming A 100 (2004), 537–568
Hall, P. [1935]: On representatives of subsets. Journal of the London Mathematical Society 10 (1935), 26–30
Halmos, P.R., and Vaughan, H.E. [1950]: The marriage problem. American Journal of Mathematics 72 (1950), 214–215
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
König, D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465
König, D. [1931]: Graphs and matrices. Matematikaiés Fizikai Lapok 38 (1931), 116–119 [in Hungarian]
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
Kuhn, H.W. [1955]: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2 (1955), 83–97
Lovász, L. [1972]: A note on factor-critical graphs. Studia Scientiarum Mathematicarum Hungarica 7 (1972), 279–280
Lovász, L. [1979]: On determinants, matchings and random algorithms. In: Fundamentals of Computation Theory (L. Budach, ed.), Akademie-Verlag, Berlin 1979, pp. 565–574
Mendelsohn, N.S., and Dulmage, A.L. [1958]: Some generalizations of the problem of distinct representatives. Canadian Journal of Mathematics 10 (1958), 230–241
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
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
Mulmuley, K., Vazirani, U.V., and Vazirani, V.V. [1987]: Matching is as easy as matrix inversion. Combinatorica 7 (1987), 105–113
Petersen, J. [1891]: Die Theorie der regulären Graphen. Acta Mathematica 15 (1891), 193–220
Rabin, M.O., and Vazirani, V.V. [1989]: Maximum matchings in general graphs through randomization. Journal of Algorithms 10 (1989), 557–567
Rizzi, R. [1998]: König’s edge coloring theorem without augmenting paths. Journal of Graph Theory 29 (1998), 87
Schrijver, A. [1998]: Counting 1-factors in regular bipartite graphs. Journal of Combinatorial Theory B 72 (1998), 122–135
Sperner, E. [1928]: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27 (1928), 544–548
Szegedy, B., and Szegedy, C. [2006]: Symplectic spaces and ear-decomposition of matroids. Combinatorica 26 (2006), 353–377
Szigeti, Z. [1996]: On a matroid defined by ear-decompositions. Combinatorica 16 (1996), 233–241
Tutte, W.T. [1947]: The factorization of linear graphs. Journal of the London Mathematical Society 22 (1947), 107–111
Vazirani, V.V. [1994]: A theory of alternating paths and blossoms for proving correctness of the \( O\left( {\sqrt V E} \right) \) general graph maximum matching algorithm. Combinatorica 14 (1994), 71–109
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(2008). Maximum Matchings. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71844-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-71844-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71843-7
Online ISBN: 978-3-540-71844-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)