The Multilevel Capacitated Minimum Spanning Tree Problem

In this paper, we consider the multilevel capacitated minimum spanning tree (MLCMST) problem, a generalization of the well-known capacitated minimum spanning tree (CMST) problem, that allows for multiple facility types in the design of the network. We develop two flow-based mixed integer programming formulations that can be used to find tight lower bounds for MLCMST problems with up to 150 nodes. We also develop several heuristic procedures for the MLCMST problem. First, we present a savings-based heuristic. Next, we develop local search algorithms that use exponential size, node-based, cyclic and path exchange neighborhoods. Finally, we develop a hybrid genetic algorithm for the MLCMST. Extensive computational results on a large set of test problems indicate that the genetic algorithm is robust and, among the heuristics, generates the best solutions. They are typically 6.09% from the lower bound and 0.25% from the optimal solution value.

[1]  P. McGregor,et al.  Network Design: An Algorithm for the Access Facility Location Problem , 1977, IEEE Trans. Commun..

[2]  R. Sharma.  Design of an Economical Multidrop Network Topology with Capacity Constraints , 1983, IEEE Trans. Commun..

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

[4]  Dushyant Sharma,et al.  Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem , 2001, Math. Program..

[5]  Oktay Günlük,et al.  Capacitated Network Design - Polyhedral Structure and Computation , 1996, INFORMS J. Comput..

[6]  Geir Dahl,et al.  A Cutting Plane Algorithm for Multicommodity Survivable Network Design Problems , 1998, INFORMS J. Comput..

[7]  Thomas L. Magnanti,et al.  Modeling and Solving the Two-Facility Capacitated Network Loading Problem , 1995, Oper. Res..

[8]  Jean-Yves Potvin,et al.  Tabu Search for a Network Loading Problem with Multiple Facilities , 2000, J. Heuristics.

[9]  L. R. Esau,et al.  On Teleprocessing System Design Part II: A Method for Approximating the Optimal Network , 1966, IBM Syst. J..

[10]  Bezalel Gavish,et al.  Topological design of telecommunication networks-local access design methods , 1991, Ann. Oper. Res..

[11]  M. Goldstein,et al.  The One-Terminal TELPAK Problem , 1971, Oper. Res..

[12]  Thomas L. Magnanti,et al.  The convex hull of two core capacitated network design problems , 1993, Math. Program..

[13]  Luís Gouveia,et al.  A comparison of directed formulations for the capacitated minimal spanning tree problem , 1993, Telecommun. Syst..

[14]  Emanuel Falkenauer,et al.  A hybrid grouping genetic algorithm for bin packing , 1996, J. Heuristics.

[15]  Michel Gendreau,et al.  A TABU SEARCH ALGORITHM FOR THE CAPACITATED SHORTEST SPANNING TREE PROBLEM. , 1997 .

[16]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[17]  Oktay Günlük,et al.  Minimum cost capacity installation for multicommodity network flows , 1998, Math. Program..

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

[19]  R. EsauL.,et al.  On teleprocessing system design , 1966 .

[20]  Stefan Voß,et al.  Capacitated minimum spanning trees: algorithms using intelligent search , 1995 .