Duality and Distance Constraints for the Nonlinear p-Center Problem and Covering Problem on a Tree Network

The problem of locating a fixed number, p, of facilities centers on a network, where there are constraints on the center locations and where the centers provide a service to customers demand points located at vertices of the network is addressed. The cost or "loss" of servicing a given demand point is a nonlinear function of the distance between the demand point and the closest center. We consider the case where the network has special structure a tree network, i.e., there is a unique shortest path between any two points on the network. We also provide and interpret a dual to this problem and give polynomially bounded procedures for solving both problems. The primal location problem is solved with the aid of a related problem for which we also give a dual.

[1]  J. Moon,et al.  Relations between packing and covering numbers of a tree , 1975 .

[2]  Gabriel Y. Handler,et al.  Finding Two-Centers of a Tree: The Continuous Case , 1978 .

[3]  N. Patel.  Locating Rural Social Service Centers in India , 1979 .

[4]  G. Handler.  Minimax Location of a Facility in an Undirected Tree Graph , 1973 .

[5]  Chihchih C. Lin.  Letter to the Editor—On Vertex Addends in Minimax Location Problems , 1975 .

[6]  George L. Nemhauser,et al.  Easy and hard bottleneck location problems , 1979, Discret. Appl. Math..

[7]  R. Chandrasekaran,et al.  Location on Tree Networks: P-Centre and n-Dispersion Problems , 1981, Math. Oper. Res..

[8]  S. Hakimi,et al.  On p -Centers in Networks , 1978 .

[9]  R. L. Francis,et al.  A Minimax Location Problem on a Network , 1974 .

[10]  Arie Tamir,et al.  Polynomially bounded algorithms for locatingp-centers on a tree , 1982, Math. Program..

[11]  S. L. Hakimi,et al.  Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph , 1964 .

[12]  S. Hakimi.  Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems , 1965 .

[13]  Peter J. Slater,et al.  Matchings and transversals in hypergraphs, domination and independence-in trees , 1979, J. Comb. Theory, Ser. B.

[14]  Arie Tamir,et al.  An O ((nlog p)2) Algorithm for the Continuous p-Center Problem on a Tree , 1980, SIAM J. Matrix Anal. Appl..

[15]  E. Minieka.  The m-Center Problem , 1970 .

[16]  Richard L. Francis,et al.  A Round-Trip Location Problem on a Tree Graph , 1976 .

[17]  Timothy J. Lowe,et al.  Convex Location Problems on Tree Networks , 1976, Oper. Res..

[18]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[19]  Richard L. Francis.  A Note on a Nonlinear Minimax Location Problem on a Tree Network. , 1977 .

[20]  A. J. Goldman.  Minimax Location of a Facility in a Network , 1972 .

[21]  Timothy J. Lowe,et al.  Distance Constraints for Tree Network Multifacility Location Problems , 1978, Oper. Res..

[22]  W. A. Horn.  Three Results for Trees, Using Mathematical Induction , 1972 .

[23]  P. M. Dearing.  Minimax location problems with nonlinear costs , 1977 .