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Up- and downgrading the 1-center in a network

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  • Gassner, Elisabeth

Abstract

We study budget constrained network improvement and degrading problems based on the vertex 1-center problem on graphs: given a graph with vertex weights and edge lengths the task is to decrease and increase the vertex weights within certain limits such that the optimal 1-center objective value with respect to the new weights is minimized and maximized, respectively. The upgrading (improvement) problem is shown to be solvable in time provided that the distance matrix is given. The downgrading 1-center problem is shown to be strongly -hard on general graphs but can be solved in time on trees. As byproduct we suggest an algorithm that solves the problem of minimizing over the upper envelope of n piecewise linear functions in time where K is the total number of breakpoints.

Suggested Citation

  • Gassner, Elisabeth, 2009. "Up- and downgrading the 1-center in a network," European Journal of Operational Research, Elsevier, vol. 198(2), pages 370-377, October.
    • Handle: RePEc:eee:ejores:v:198:y:2009:i:2:p:370-377
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    References listed on IDEAS

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    1. Sven O. Krumke & Madhav V. Marathe & Hartmut Noltemeier & R. Ravi & S. S. Ravi, 1998. "Approximation Algorithms for Certain Network Improvement Problems," Journal of Combinatorial Optimization, Springer, vol. 2(3), pages 257-288, September.
    2. Zhang, J. Z. & Yang, X. G. & Cai, M. C., 2004. "Inapproximability and a polynomially solvable special case of a network improvement problem," European Journal of Operational Research, Elsevier, vol. 155(1), pages 251-257, May.
    3. Burkard, Rainer E. & Lin, Yixun & Zhang, Jianzhong, 2004. "Weight reduction problems with certain bottleneck objectives," European Journal of Operational Research, Elsevier, vol. 153(1), pages 191-199, February.
    4. Elisabeth Gassner, 2008. "The inverse 1-maxian problem with edge length modification," Journal of Combinatorial Optimization, Springer, vol. 16(1), pages 50-67, July.
    5. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
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    Cited by:

    1. Kien Trung Nguyen & Ali Reza Sepasian, 2016. "The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 872-884, October.
    2. Frank Plastria, 2016. "Up- and downgrading the euclidean 1-median problem and knapsack Voronoi diagrams," Annals of Operations Research, Springer, vol. 246(1), pages 227-251, November.
    3. Nguyen, Kien Trung & Hung, Nguyen Thanh, 2021. "The minmax regret inverse maximum weight problem," Applied Mathematics and Computation, Elsevier, vol. 407(C).
    4. Baldomero-Naranjo, Marta & Kalcsics, Jörg & Marín, Alfredo & Rodríguez-Chía, Antonio M., 2022. "Upgrading edges in the maximal covering location problem," European Journal of Operational Research, Elsevier, vol. 303(1), pages 14-36.
    5. Zhi-Ming Chen & Cheng-Hsiung Lee & Hung-Lin Lai, 2022. "Speedup the optimization of maximal closure of a node-weighted directed acyclic graph," OPSEARCH, Springer;Operational Research Society of India, vol. 59(4), pages 1413-1437, December.
    6. Elisabeth Gassner, 2009. "A game-theoretic approach for downgrading the 1-median in the plane with Manhattan metric," Annals of Operations Research, Springer, vol. 172(1), pages 393-404, November.

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