# The economic addition of functionality to a network

## Abstract

In the operation of communication and computer networks, it may become desirable or necessary to add a new function to the network through the placement of the corresponding electronic device within certain existing user locations. This will involve deciding which user locations will have devices placed at them as well as deciding an assignment of users to device locations. The objective when adding the new function is to choose these locations and assignments such that the combined cost of placing the devices and routing users to their assigned device locations is minimized. This problem, which we call the device placement problem, is closely related to the simple plant location problem and the *p*-median problem. Like these problems, the device placement problem is NP-hard, and thus it is highly unlikely that efficient methods for solving this problem to optimality exist.

We discuss and test several heuristic methods for the device placement problem, as well as a very efficient method for obtaining lower bounds. Our methods are based on relating the problem to a certain restricted spanning tree problem, which leads to conceptually easy ideas.

## Keywords

facility location concentrator location simple plant location network design restricted spanning trees## Preview

Unable to display preview. Download preview PDF.

## References

- 1.N.R. Achuthan and L. Caccetta. Minimum weight spanning trees with bounded diameter.
*Australasian Journal of Combinatorics*, 5:261–276, 1992.Google Scholar - 2.N.R. Achuthan, L. Caccetta, P. Caccetta, and J.F. Geelen. Computational methods for the diameter restricted minimum weight spanning tree problem.
*Australasian Journal of Combinatorics*, 10:51–71, 1994.Google Scholar - 3.A. Balakrishnan, T.L. Magnanti, A. Shulman, and R.T. Wong. Models for planning capacity expansion in local access telecommunication networks.
*Annals of Operations Research*, 33:239–284, 1991.Google Scholar - 4.O. Bilde and J. Krarup. Sharp lower bounds and efficient algorithms for the simple plant location problem.
*Annals of Discrete Mathematics*, 1:79–97, 1977.Google Scholar - 5.M.L. Brandeau and S.S. Chiu. An overview of representative problems in location research.
*Management Science*, 35:645–674, 1989.Google Scholar - 6.P.M. Camerini, A. Colorni, and F. Maffioli. Some experience in applying a stochastic method to location problems.
*Mathematical Programming Study*, 26:229–232, 1986.Google Scholar - 7.G. Cornuejols, M. Fisher, and G.L. Nemhauser. On the uncapacitated location problem.
*Annals of Discrete Mathematics*, 1:163–177, 1977.Google Scholar - 8.G. Cornuejols, G.L. Nemhauser, and L.A. Wolsey. The uncapacited facility location problem. In R.L. Francis and P. Mirchandani, editors,
*Discrete Location Theory*. Wiley Interscience, 1990.Google Scholar - 9.M.A. Efroymson and T.L. Ray. A branch-bound algorithm for plant location.
*Operations Research*, 14:361–368, 1966.Google Scholar - 10.D. Erlenkotter. A dual-based procedure for uncapacitated facility location.
*Operations Research*, 26:992–1009, 1978.Google Scholar - 11.M.R. Garey and D.S. Johnson.
*Computers and Intractability — A guide to the Theory of NP-completeness*. Freeman, 1979.Google Scholar - 12.O. Kariv and S.L. Hakimi. An algorithmic approach to network location problem. II: The
*p*-medians.*SIAM Journal on Applied Mathematics*, 37:539–560, 1979.Google Scholar - 13.J. Krarup and P.M. Pruzan. The simple plant location problem: Survey and synthesis.
*European Journal of Operations Research*, 12:36–81, 1983.Google Scholar - 14.J.B. Jr. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem.
*Proceedings of the AMS*, 7:48–50, 1956.Google Scholar - 15.P.V. McGregor and D. Shen. Network design: An algorithm for the facility location problem.
*IEEE Transactions on Communications*, Com-25:61–73, 1977.Google Scholar - 16.G.L. Narula, U.I. Ogbu, and H.M. Samuelson. An algorithm for the p-median problem.
*Operations Research*, 25:709–713, 1977.Google Scholar - 17.C.H. Papadimitriou and M. Yannakakis. The complexity of restricted spanning tree problems.
*Journal of the ACM*, 29:285–309, 1982.Google Scholar - 18.R.C. Prim. Shortest connection networks and some generalizations.
*Bell System Technical Journal*, 36:1389–1401, 1957.Google Scholar - 19.G.M. Schneider and M.N. Zastrow. An algorithm for the design of multilevel concentrator networks.
*Computer Networks*, 6:1–11, 1982.Google Scholar - 20.K. Spielberg. Algorithms for the simple plant-location problem with some side conditions.
*Operations Research*, 17:85–111, 1969.Google Scholar - 21.B.C. Tansel, R.L. Francis, and T.J Lowe. Location on Networks: a survey. Part I: the
*p*-center and*p*-median problems.*Management Science*, 29:482–511, 1983.Google Scholar