Finding minimum cost directed trees with demands and capacities

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.

[1]  Michel Minoux,et al.  Networks synthesis and optimum network design problems: Models, solution methods and applications , 1989, Networks.

[2]  Bezalel Gavish,et al.  Topological design of centralized computer networks - formulations and algorithms , 1982, Networks.

[3]  Christos H. Papadimitriou,et al.  The complexity of the capacitated tree problem , 1978, Networks.

[4]  Bezalel Gavish,et al.  Formulations and Algorithms for the Capacitated Minimal Directed Tree Problem , 1983, JACM.

[5]  Michel Minoux,et al.  Network Synthesis and Dynamic Network Optimization , 1987 .

[6]  T. Gönen,et al.  Review of distribution system planning models: a model for optimal multistage planning , 1986 .

[7]  John E. Beasley.  An algorithm for the steiner problem in graphs , 1984, Networks.

[8]  Ellis L. Johnson,et al.  Solving Large-Scale Zero-One Linear Programming Problems , 1983, Oper. Res..

[9]  Yash P. Aneja,et al.  An integer linear programming approach to the steiner problem in graphs , 1980, Networks.

[10]  R. N. Adams,et al.  Optimal planning of power networks using mixed-integer programming. Part 1: Static and time-phased network synthesis , 1974 .

[11]  Laurence A. Wolsey,et al.  Packing and covering a tree by subtrees , 1986, Comb..

[12]  Laurence A. Wolsey,et al.  Solving Mixed Integer Programming Problems Using Automatic Reformulation , 1987, Oper. Res..

[13]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..