Abstract
The convex ordered median problem is a generalization of the median, the k-centrum or the center problem. The task of the associated inverse problem is to change edge lengths at minimum cost such that a given vertex becomes an optimal solution of the location problem, i.e., an ordered median. It is shown that the problem is NP-hard even if the underlying network is a tree and the ordered median problem is convex and either the vertex weights are all equal to 1 or the underlying problem is the k-centrum problem. For the special case of the inverse unit weight k-centrum problem a polynomial time algorithm is developed.
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References
Burkard RE, Pleschiutschnig C, Zhang J (2004) Inverse median problems. Discrete Optim 1(1):23–39
Burkard RE, Pleschiutschnig C, Zhang J (2008a) The inverse 1-median problem on a cycle. Discrete Optim 5(2):242–253
Burkard RE, Galavii M, Gassner E (2008b) The inverse Fermat-Weber problem. Technical Report 2008-14, Graz University of Technology. http://www.math.tugraz.at/fosp/pdfs/tugraz_0107.pdf
Cai MC, Yang XG, Zhang JZ (1999) The complexity analysis of the inverse center location problem. J Glob Optim 15(2):213–218
Dominguez-Marin P (2003) The discrete ordered median problem: models and solution methods. Combinatorial optimization series, vol 15. Springer, Berlin
Garey MR, Johnson DS (1979) Computers and intractability. A guide to the theory of NP-completeness. A series of books in the mathematical sciences. Freeman, New York
Gassner E (2008) The inverse 1-maxian problem with edge length modification. J Comb Optim 16(1):50–67
Goldman AJ (1971) Optimal center location in simple networks. Transp Sci 5:212–221
Heuberger C (2004) Inverse combinatorial optimization: a survey on problems, methods, and results. J Comb Optim 8(3):329–361
Kalcsics J, Nickel S, Puerto J, Tamir A (2002) Algorithmic results for ordered median problems. Oper Res Lett 30(3):149–158
Nickel S, Puerto J (1999) A unified approach to network location problems. Networks 34(4):283–290
Nickel S, Puerto J (2005) Location theory—a unified approach. Springer, Berlin
Tamir A (2001) The k-centrum multi-facility location problem. Discrete Appl Math 109(3):293–307
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This research has been supported by the Austrian Science Fund (FWF) Project P18918-N18.
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Gassner, E. An inverse approach to convex ordered median problems in trees. J Comb Optim 23, 261–273 (2012). https://doi.org/10.1007/s10878-010-9353-3
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DOI: https://doi.org/10.1007/s10878-010-9353-3