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.(n3)-implementation.
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References
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
Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, Chapter 11
Pulleyblank, W.R. [1995]: Matchings and extensions. In: Handbook of Combinatorics; Vol. 1 (R.L. Graham, M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam 1995
Balinski, M.L. [1972]: Establishing the matching polytope. Journal of Combinatorial Theory 13 (1972), 1–13
Ball, M.O., and Derigs, U. [1983]: An analysis of alternative strategies for implementing matching algorithms. Networks 13 (1983), 517–549
Birkhoff, G. [1946]: Tres observaciones sobre el algebra lineal. Revista Universidad Nacional de Tucumán, Series A 5 (1946), 147–151
Burkard, R., Dell’Amico, M., and Martello, S. [2009]: Assignment Problems. SIAM, Philadelphia 2009
Cook, W., and Rohe, A. [1999]: Computing minimum-weight perfect matchings. INFORMS Journal of Computing 11 (1999), 138–148
Cunningham, W.H., and Marsh, A.B. [1978]: A primal algorithm for optimum matching. Mathematical Programming Study 8 (1978), 50–72
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
Egerváry, E. [1931]: Matrixok kombinatorikus tulajdonságairol. Matematikai és Fizikai Lapok 38 (1931), 16–28 [in Hungarian]
Gabow, H.N. [1973]: Implementation of algorithms for maximum matching on non-bipartite graphs. Ph.D. Thesis, Stanford University, Dept. of Computer Science, 1973
Gabow, H.N. [1976]: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. Journal of the ACM 23 (1976), 221–234
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
Grötschel, M., and Pulleyblank, W.R. [1981]: Weakly bipartite graphs and the max-cut problem. Operations Research Letters 1 (1981), 23–27
Kolmogorov, V. [2009]: Blossom V: a new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation 1 (2009), 43–67
Kuhn, H.W. [1955]: The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2 (1955), 83–97
Lipton, R.J., and Tarjan, R.E. [1979]: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36 (1979), 177–189
Lipton, R.J., and Tarjan, R.E. [1980]: Applications of a planar separator theorem. SIAM Journal on Computing 9 (1980), 615–627
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
Mehlhorn, K., and Schäfer, G. [2000]: Implementation of O(nmlogn) 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; also electronically in The ACM Journal of Experimental Algorithmics 7 (2002)
Monge, G. [1784]: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences 2 (1784), 666–704
Munkres, J. [1957]: Algorithms for the assignment and transportation problems. Journal of the Society for Industrial and Applied Mathematics 5 (1957), 32–38
Naddef, D., and Pulleyblank, W.R. [1981]: Matchings in regular graphs. Discrete Mathematics 34 (1981), 283–291
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
Schrijver, A. [1983a]: Short proofs on the matching polyhedron. Journal of Combinatorial Theory B 34 (1983), 104–108
Schrijver, A. [1983b]: Min-max results in combinatorial optimization. In: Mathematical Programming; The State of the Art – Bonn 1982 (A. Bachem, M. Grötschel, B. Korte, eds.), Springer, Berlin 1983, pp. 439–500
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
Weber, G.M. [1981]: Sensitivity analysis of optimal matchings. Networks 11 (1981), 41–56
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Korte, B., Vygen, J. (2012). Weighted Matching. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24488-9_11
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DOI: https://doi.org/10.1007/978-3-642-24488-9_11
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