Abstract
Given a directed graph, we consider the problem of finding a rooted directed tree (or branching) satisfying given demands at all the nodes and capacity constraints on the arcs. Various integer programming formulations are compared, including flow and multicommodity flow formulations and two partitioning-type formulations involving directed subtrees. Computational results concerning an application to the design of a low voltage electricity network are given. For the class of problems considered, one of the partitioning formulations allows us to solve problems with up to 100 nodes and several hundred arcs.
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References
R.N. Adam and M.A. Laughton, Optimal planning of power networks using mixed integer programming, Proc. IEE 121(1974)139–147.
Y.P. Aneja, An integer linear programming approach to the Steiner problem in graphs, Networks 10(1980)167–178.
A. Balakrishnan, T.L. Magnanti and R.T. Wong, A network design problem,Oberwolfach Conf. on Applications of Combinatorial Optimization (1989).
M.O. Ball, W.G. Liu and W.R. Pulleyblank, Two terminal Steiner tree polyhedra, in:Contributions to Operations Research and Economics: The 20th Anniversary of CORE, ed. B. Cornet and H. Tulkens (MIT Press, 1990).
I. Barany, J. Edmonds and L.A. Wolsey, Packing and covering a tree by subtrees, Combinatorica 6(1986)221–233.
J.E. Beasley, An algorithm for the Steiner problem in graphs, Networks 14(1984)147–160.
Ch. Bousba, Planification des réseaux électriques de distribution: une approache par la programmation mathématique en nombres entiers, Thèse de Doctorat en Sciences Appliquées, Université de Louvain, Louvain-la-Neuve, Belgium (1989).
H. Crowder, E.L. Johnson and M.W. Padberg, Solving large-scale zero-one linear programming problems, Oper. Res. 31(1983)803–834.
B. Gavish, Topological design of centralized computer networks. Formulations and algorithms, Networks 12(1982)355–377.
B. Gavish, Formulations and algorithms for the capacitated minimal directed tree problem, J. ACM 30(1983)118–132.
B. Gavish, Augmented Lagrangian based algorithms for centralized network design, Working Paper 8321, Graduate School of Management, University of Rochester.
T. Gonen and I.J. Ramirez-Rosado, Review of distribution system planning models: A model for optimal multistage planning, IEE Proc. 133(C)(1986)397–408.
M. Minoux, Network synthesis and dynamic network optimization, Ann. Discr. Math. 31(1987)283–324.
M. Minoux, Network synthesis and optimum network design problems, Networks 19(1989)313–360.
Ch. Papadimitriou, The complexity of the capacitated tree problem, Networks 8(1978)217–230.
A. Prodon, T.M. Liebling and H. Gröflin, Steiner's problem on two-trees, Technical Report RO 850315, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne (1985).
T.J. Van Roy and L.A. Wolsey, Solving mixed integer programs by automatic reformulation, Oper. Res. 35(1987)45–57.
L.A. Wolsey, Valid inequalities for knapsack problems with generalized upper bounds, Technical Report RO 870630, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne (1987).
R.T. Wong, A dual ascent approach for Steiner tree problems on a directed graph, Math. Progr. 28(1984)271–287.
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The research of the first author was partially supported by PAC Contract No. 87/92-106 for computation.
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Bousba, C., Wolsey, L.A. Finding minimum cost directed trees with demands and capacities. Ann Oper Res 33, 285–303 (1991). https://doi.org/10.1007/BF02071977
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DOI: https://doi.org/10.1007/BF02071977