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An inverse approach to convex ordered median problems in trees

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  • Elisabeth Gassner

    (Technische Universität Graz)

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.

Suggested Citation

  • Elisabeth Gassner, 2012. "An inverse approach to convex ordered median problems in trees," Journal of Combinatorial Optimization, Springer, vol. 23(2), pages 261-273, February.
    • Handle: RePEc:spr:jcomop:v:23:y:2012:i:2:d:10.1007_s10878-010-9353-3
    DOI: 10.1007/s10878-010-9353-3
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    References listed on IDEAS

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    1. A. J. Goldman, 1971. "Optimal Center Location in Simple Networks," Transportation Science, INFORMS, vol. 5(2), pages 212-221, May.
    2. Elisabeth Gassner, 2008. "The inverse 1-maxian problem with edge length modification," Journal of Combinatorial Optimization, Springer, vol. 16(1), pages 50-67, July.
    3. 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 & Nguyen Thanh Hung, 2020. "The inverse connected p-median problem on block graphs under various cost functions," Annals of Operations Research, Springer, vol. 292(1), pages 97-112, September.
    2. Kien Nguyen & Lam Anh, 2015. "Inverse $$k$$ k -centrum problem on trees with variable vertex weights," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(1), pages 19-30, August.
    3. Nguyen, Kien Trung & Chassein, André, 2015. "The inverse convex ordered 1-median problem on trees under Chebyshev norm and Hamming distance," European Journal of Operational Research, Elsevier, vol. 247(3), pages 774-781.
    4. Esmaeil Afrashteh & Behrooz Alizadeh & Fahimeh Baroughi & Kien Trung Nguyen, 2018. "Linear Time Optimal Approaches for Max-Profit Inverse 1-Median Location Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(05), pages 1-22, October.
    5. 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.
    6. Kien Trung Nguyen & Huong Nguyen-Thu & Nguyen Thanh Hung, 2018. "On the complexity of inverse convex ordered 1-median problem on the plane and on tree networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 147-159, October.
    7. Kien Trung Nguyen, 2019. "The inverse 1-center problem on cycles with variable edge lengths," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(1), pages 263-274, March.
    8. Alizadeh, Behrooz & Afrashteh, Esmaeil, 2020. "Budget-constrained inverse median facility location problem on tree networks," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    9. Le Xuan Dai & Kien Trung Nguyen & Le Phuong Thao & Pham Thi Vui, 2024. "Some robust inverse median problems on trees with interval costs," Computational Management Science, Springer, vol. 21(2), pages 1-25, December.
    10. Kien Trung Nguyen, 2016. "Inverse 1-Median Problem on Block Graphs with Variable Vertex Weights," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 944-957, March.
    11. Behrooz Alizadeh & Rainer Burkard, 2013. "A linear time algorithm for inverse obnoxious center location problems on networks," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(3), pages 585-594, September.

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