Abstract
This paper addresses the problem of optimally modifying the edge lengths such that a prespecified vertex becomes the furthest vertex from a given fixed vertex in the perturbed network. We call this problem the inverse eccentric vertex problem. We show that the problem is \(NP\)-complete even on cactus graphs. However, if the underlying graph is a cycle or a tree, we develop efficient algorithms with linear time complexity.
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References
Ahuja RK, Orlin JB (2002) Combinatorial algorithms for inverse network flow problems. Networks 40: 181–187
Ahuja RK, Orlin JB (2001) Inverse optimization. Oper Res 49:771–783
Alizadeh B, Burkard RE (2011a) Combinatorial algorithms for inverse absolute and vertex 1-center location problems on trees. Networks 58:190–200
Alizadeh B, Burkard RE (2011b) 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
Bezad M, Simpson JE (1976) Eccentric sequences and eccentric sets in graphs. Discret Math 16:178–193
Burkard RE, Pleschiutschnig C, Zhang JZ (2004) Inverse median problems. Discret Optim 1:23–39
Burton D (1993) Inverse shortest path problem. PhD thesis
Cai MC, Yang XG, Zhang JZ (1999) The complexity analysis of the inverse center location problem. J Glob Optim 15:213–218
Garey MR, Johnson DS (1979) Computers and intractability, a guide to the theory of \(NP\)-completeness. W. H. Freeman and Co, New York
Heuberger C (2004) Inverse combinatorial optimization: a survey on problems, methods, and results. J Comb Optim 8:329–361
Hochbaum DS (2003) Efficient algorithms for the inverse spanning-tree problem. Oper Res 51:785–797
Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems, I: the p-centers. SIAM J Appl Math 37:513–538
Megiddo N (1983) Linear-time algorithms for linear programming in \(\mathbb{R}^{3}\) and related problems. SIAM J Comput 12:759–776
Mneimneh M, Sakallah K (2003) Computing vertex eccentricity in exponentially large graphs: QBF formulation and solution. In: Proceedings of 6th international conference SAT03, volume 2919 of LNCS
Reid KB, Gu W (1992) Peripheral and eccentric vertices in graphs. Gr Comb 8:361–375
Acknowledgments
The authors would like to thank anonymous referees for helpful feedback that helped to improve the paper. This work was financial supported by DAAD (German Academic Exchange Service).
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Nguyen, K.T., Chassein, A. Inverse eccentric vertex problem on networks. Cent Eur J Oper Res 23, 687–698 (2015). https://doi.org/10.1007/s10100-014-0367-2
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DOI: https://doi.org/10.1007/s10100-014-0367-2