Abstract
In this chapter we introduce two more combinatorial optimization problems, the Minimum Weight b-Matching Problem in Section 12.1 and the Minimum Weight T-Join Problem in Section 12.2. Both can be regarded as generalizations of the Minimum Weight Perfect Matching Problem and also include other important problems. On the other hand, both problems can be reduced to the Minimum Weight Perfect Matching Problem. They have combinatorial polynomial-time algorithms as well as polyhedral descriptions. Since in both cases the Separation Problem turns out to be solvable in polynomial time, we obtain another polynomial-time algorithm for the general matching problems (using the Ellipsoid Method; see Section 4.6). In fact, the Separation Problem can be reduced to finding a minimum capacity T-cut in both cases; see Sections 12.3 and 12.4. This problem, finding a minimum capacity cut δ(X) such that |X ∩ T| is odd for a specified vertex set T, can be solved with network flow techniques.
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References
General Literature
Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., and Schrijver, A. [1998]: Combinatorial Optimization. Wiley, New York 1998, Sections 5.4 and 5.5
Frank, A. [1996]: A survey on T-joins, T-cuts, and conservative weightings. In: Combinatorics, Paul Erdős is Eighty; Volume 2 (D. Miklós, V.T. Sós, T. Szőnyi, eds.), Bolyai Society, Budapest 1996, pp. 213–252
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
Lovász, L., and Plummer, M.D. [1986]: Matching Theory. Akadémi ai Kiadó, Budapest 1986, and North-Holland, Amsterdam 1986
Schrijver, A. [1983]: Min-max results in combinatorial optimization; Section 6. In: Mathematical Programming; The State of the Art — Bonn 1982 (A. Bachern, M. Grötschel, B. Korte, eds.), Springer, Berlin 1983, pp. 439–500
Cited References
Anstee, R.P. [1987]: A polynomial algorithm for b-matchings: an alternative approach. Information Processing Letters 24 (1987), 153–157
Caprara, A., and Fischetti, M. [1996]: 0, ½-Chaátal-Gomory cuts. Mathematical Programming 74(3) (1996)
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
Edmonds, J., and Johnson, E.L. [1970]: Matching: A well-solved class of integer linear programs. In: Combinatorial Structures and Their Applications; Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications 1969 (R. Guy, H. Hanani, N. Sauer, J. Schonheim, eds.), Gordon and Breach, New York 1970, pp. 69–87
Edmonds, J., and Johnson, E.L. [1973]: Matching, Euler tours and the Chinese postman problem. Mathematical Programming 5 (1973), 88–124
Guan, M. [1962]: Graphic programming using odd and even points. Chinese Mathematics 1 (1962), 273–277
Hadlock, F. [1975]: Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4 (1975), 221–225
Marsh, A.B. [1979]: Matching algorithms. Ph.D. thesis, Johns Hopkins University, Baltimore 1979
Padberg, M.W., and Rao, M.R. [1982]: Odd minimum cut-sets and b-matchings. Mathematics of Operations Research 7 (1982), 67–80
Pulleyblank, W.R. [1973]: Faces of matching polyhedra. Ph.D. thesis, University of Waterloo, 1973
Pulleyblank, W.R. [1980]: Dual integrality in b-matching problems. Mathematical Programming Study 12 (1980), 176–196
Sebő, A. [1987]: A quick proof of Seymour’s theorem on T-joins. Discrete Mathematics 64 (1987), 101–103
Seymour, P.D. [1981]: On odd cuts and multicommodity flows. Proceedings of the London Mathematical Society (3) 42 (1981), 178–192
Tutte, W.T. [1952]: The factors of graphs. Canadian Journal of Mathematics 4 (1952), 314–328
Tutte, W.T. [1954]: A short proof of the factor theorem for finite graphs. Canadian Journal of Mathematics 6 (1954), 347–352
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© 2002 Springer-Verlag Berlin Heidelberg
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Korte, B., Vygen, J. (2002). b-Matchings and T-Joins. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21711-5_12
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DOI: https://doi.org/10.1007/978-3-662-21711-5_12
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