Abstract
The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l ∞ norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is \({\mathcal{NP}}\) -hard.
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References
Alizadeh B., Burkard R.E., Pferschy U.: Inverse 1-center location problems with edge length augmentation on trees. Computing 86(4), 331–343 (2009)
Balas E., Zemel E.: An algorithm for large zero-one knapsack problems. Oper. Res. 28, 1130–1154 (1980)
Bonab F.B., Burkard R.E., Alizadeh B.: Inverse median location problems with variable coordinates. Cen. Eur. J. Oper. Res. 18(3), 365–381 (2010)
Burkard R.E., Galavii M., Gassner E.: The inverse Fermat-Weber problem. Eur. J. Oper. Res. 206(1), 11–17 (2010)
Burkard R.E., Pleschiutschnig C., Zhang J.: Inverse median problems. Discrete Optim. 1, 23–39 (2004)
Burkard R.E., Pleschiutschnig C., Zhang J.Z.: The inverse 1-median problem on a cycle. Discrete Optim. 5, 242–253 (2008)
Cai M., Yang X., Zhang J.: The complexity analysis of the inverse center location problem. J. Glob. Optim. 5, 213–218 (1999)
Cao, Y.B., Guan, X.C.: A class of constrained inverse bottleneck optimization problems under weighted Hamming distance. In: Proceedings of International Joint Conference on Computational Sciences and Optimization, IEEE Computer Society vol. 2, pp. 859–863 (2009)
Duin C.W., Volgenant A.: Some inverse optimization problems under the Hamming distance. Eur. J. Oper. Res. 170, 887–899 (2006)
Galavii, M.: Inverse 1-median problems. Ph.D. Thesis, Institute of Optimization and Discrete Mathematics, Graz University of Technology, Graz (2008)
Gassner E.: The inverse 1-Maxian problem with edge length modiffication. J. Comb. Optim. 16, 50–67 (2008)
Goldman A.J.: Optimal center location in simple networks. Transp. Sci. 2, 77–91 (1962)
Guan, X.C., Zhang, B.W.: Inverse 1-median problem on trees under weighted l ∞ norm. Lect. Notes Comput. Sci. 6124, 150–160 (2010)
Guan X.C., Zhang J.Z.: Inverse constrained bottleneck problems under weighted l ∞ norm. Comput. Oper. Res. 34, 3243–3254 (2007)
Guan, X.C., Zhang, J.Z.: Inverse bottleneck optimization problems under weighted Hamming distance. Lect. Notes Comput. Sci. 4041, 220–230 (2006)
Hua et al.: Applications of mathematical models to wheat harvesting. Acta Mathematica Sinica 11, 63C75 (1961) (in Chinese) (English translation in Chinese Math. 2, 77C91 (1962))
Pruhs K., Woeginger G.J.: Approximation schemes for a class of subset selection problems. Theor. Comput. Sci. 382, 151–156 (2007)
Silvano M., Toth P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York, NY, USA (1990)
Yang X., Zhang J.: Some inverse min–max network problems under weighted l 1 and l ∞ norms with bound constraints on changes. J. Comb. Optim. 13, 123–135 (2007)
Yang X., Zhang J.: Inverse center location problem on a tree. J. Syst. Sci. Complex. 21, 651–664 (2008)
Zhang B., Zhang J., He Y.: Constrained inverse minimum spanning tree problems under the bottleneck-type Hamming distance. J. Glob. Optim. 34, 467–474 (2006)
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Guan, X., Zhang, B. Inverse 1-median problem on trees under weighted Hamming distance. J Glob Optim 54, 75–82 (2012). https://doi.org/10.1007/s10898-011-9742-x
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DOI: https://doi.org/10.1007/s10898-011-9742-x