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An optimal column-generation-with-ranking algorithm for very large scale set partitioning problems in traffic assignment. (English) Zbl 0679.90043

The paper is devoted to a method of solving some very large-scale set partitioning problems with special structure, such as the matrix decomposition problem. P. C. Gilmore and R. E. Gomory [Oper. Res. 9, 849-859 (1961; Zbl 0096.355)] applied the column generation technique to a class of combinatorial problems to obtain good approximate solutions and lower bounds. In this paper the authors show how the branch-and-bound method can be combined with the column generation technique to ensure obtaining the exact optimal integer solution. They also report extensive computational results.
Reviewer: X.Zhang

MSC:

90C10 Integer programming
90C27 Combinatorial optimization
65K05 Numerical mathematical programming methods
90C06 Large-scale problems in mathematical programming
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)

Citations:

Zbl 0096.355
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References:

[1] Balas, E.; Landweer, P. R., Traffic assignment in communication satellites, Operations Research Letters, 2, 141-147 (1983) · Zbl 0526.90065
[2] Brandt, E., Optimisation de ressources pour des transmissions par satellites utilisant l’ AMRT, (Thèse de 3éme Cycle (1982), Université de Paris-Sud Orsay: Université de Paris-Sud Orsay Paris)
[3] Busacker, R. G.; Gowen, P. J., A procedure for determining a family of minimal cost network flow patterns, (O.R.O. Technical Report 15 (1961), Johns Hopkins University)
[4] Camerini, P.; Maffioli, F.; Tartara, G., Some scheduling algorithms for SS/TDMA systems, (Proceedings of the V International Conference on Digital Satellite Communications. Proceedings of the V International Conference on Digital Satellite Communications, Genoa (1981)), 405-409
[5] Dantzig, G. B.; Orden, A.; Wolfe, P., The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific Journal of Mathematics, 5, 183-195 (1955) · Zbl 0064.39402
[6] Edmonds, J.; Karp, R. M., Theoretical improvements in algorithmic efficiency for network flow problems, Journal of the ACM, 19, 248-264 (1972) · Zbl 0318.90024
[7] Gilmore, P. C.; Gomory, R. E., A linear programming apporach to the cutting-stock problem, Operations Research, 9, 849-859 (1961) · Zbl 0096.35501
[8] Gopal, I. S., Scheduling algorithms for multi-beam communication satellites, (Ph.D. Dissertation (1982), Columbia University: Columbia University New York)
[9] Gopal, I. S.; Wong, C. K., Minimizing the number of switching in an SS/TDMA system, (Proceedings of the International Symposium on Satellite and Computer Communications. Proceedings of the International Symposium on Satellite and Computer Communications, Versailles (1983))
[10] Inukai, T., Comments on ‘Analysis of a switch matrix for an SS/TDMA systems’, (Proceedings of the IEEE, 66 (1978)), 1669-1670
[11] Inukai, T., An efficient SS/TDMA time slot assignment algorithm, IEEE Transactions on Communications, 27, 1449-1455 (1979)
[12] Ito, Y.; Urano, Y.; Muratani, T.; Yamaguchi, M., Analysis of a switch matrix for an SS?TDMA system, (Proceedings of the IEEE, 65 (1977)), 411-419
[13] Lawler, E. L., Procedure for computing the \(k\)-best solutions to discrete optimization problems and its applications to shortest path problem, Management Science, 18, 401-405 (1972) · Zbl 0234.90050
[14] Minoux, M., Optimal traffic assignment in an SS/TDMA frame: A new approach by set covering and column generation, RAIRO Recherche Opérationnelle, 20, 1-13 (1986) · Zbl 0608.90076
[15] Minoux, M., A class of combinatorial problems with polynomially solvable large scale scale set covering/set partitioning relaxations, RAIRO Recherche Opérationnelle, 21, 105-136 (1987) · Zbl 0644.90061
[16] Murty, K. G., An algorithm for ranking all the assignments in order of increasing cost, Operations Research, 16, 682-687 (1968) · Zbl 0214.18705
[17] Murty, K. G., Linear and Combinatorial Programming (1976), Wiley: Wiley New York · Zbl 0634.90037
[18] Rendl, F., On the complexity of decomposing matrices arising in satellite communications, Operations Research Letters, 4, 5-8 (1985) · Zbl 0569.90064
[19] Rendl, F. (1986), Private Communication.; Rendl, F. (1986), Private Communication.
[20] Vismara, L., Piano di accesso dei messaggi in sistemi SS/TDMA: un metodo di enumerazione implicita per minimizzare il tempo di transmissione (1982), Tesi di Laurea, Politecnico di Milano, Dipartimento di Elettronica, (Paolo Camerini i Francesco Maffioli, relatori)
[21] Yen, J. Y., Finding the \(k\)-shortest loopless paths in a network, Management Science, 17, 712-716 (1971) · Zbl 0218.90063
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