Abstract
This paper addresses the problem of modifying the edge lengths of a tree in minimum total cost such that a prespecified vertex becomes the 1-center of the perturbed tree. This problem is called the inverse 1-center problem on trees. We focus on the problem under Chebyshev norm and Hamming distance. From special properties of the objective functions, we can develop combinatorial algorithms to solve the problem. Precisely, if there does not exist any vertex coinciding with the prespecified vertex during the modification of edge lengths, the problem under Chebyshev norm or bottleneck Hamming distance is solvable in \(O(n\log n)\) time, where \(n+1\) is the number of vertices of the tree. Dropping this condition, the problem can be solved in \(O(n^{2})\) time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh B, Burkard Rainer E (2011) Combinatorial algorithms for inverse absolute and vertex 1-center location problems on trees. Networks 58(3):190–200
Alizadeh B, Burkard RE (2011) Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees. Discret Appl Math 159:706–716
Alizadeh B, Burkard RE, Pferschy U (2009) Inverse 1-center location problems with edge length augmentation on trees. Computing 86:331–343
Bonab FB, Burkard RE, Gassner E (2011) Inverse p-median problems with variable edge lengths. Math Methods Oper Res 73:263–280
Burkard RE, Pleschiutschnig C, Zhang J (2004) Inverse median problems. Discret Optim 1:23–39
Burkard RE, Pleschiutschnig C, Zhang J (2008) The inverse 1-median problem on a cycle. Discret Optim 5:242–253
Cai MC, Yang XG, Zhang JZ (1999) The complexity analysis of inverse center location problem. J Glob Optim 15:213–218
Drezner Z, Hamacher HW (2004) Facility location, applications and theory. Springer, Berlin
Gassner E (2009) Up- and downgrading the 1-center in a network. Eur J Oper Res 198:370377
Galavii M (2010) The inverse 1-median problem on a tree and on a path. Electron Notes Discret Math 36:1241–1248
Guan X, Zhang B (2012) Inverse 1-median problem on trees under weighted Hamming distance. J Glob Optim 54:75–82
Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems. Part I: p-centers. Siam J Appl Math 37:539–560
Love RF, Morris JG, Wesolowsky GO (1988) Facilities location: models and methods. Elsevier Science Publishing Co., New York
Nguyen KT (2014) Inverse location theory with ordered median function and other extensions. PhD Thesis, Kaiserslautern University, Germany
Nguyen KT, Anh LQ (2015) Inverse \(k\)-centrum problem on trees with variable vertex weights. Math Meth Oper Res. doi:10.1007/s00186-015-0502-4
Nguyen KT (2015) Reverse 1-center problem on weighted trees. Optimization. doi:10.1080/02331934.2014.994626
Nguyen KT, Chassein A (2014) Inverse eccentric vertex problem on networks. Cent Eur J Oper Res. doi:10.1007/s10100-014-0367-2
Acknowledgments
The authors are grateful to the anonymous referees for valuable comments which help to improve this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nguyen, K.T., Sepasian, A.R. The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance. J Comb Optim 32, 872–884 (2016). https://doi.org/10.1007/s10878-015-9907-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-015-9907-5