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Inverse 1-Median Problem on Block Graphs with Variable Vertex Weights

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  • Kien Trung Nguyen

    (Cantho University)

Abstract

This paper addresses the problem of modifying the vertex weights of a block graph at minimum total cost so that a prespecified vertex becomes a 1-median of the perturbed graph. We call this problem the inverse 1-median problem on block graphs with variable vertex weights. For block graphs with equal edge lengths in each block, we can formulate the problem as a univariate optimization problem. By the convexity of the objective function, the local optimizer is also the global one. Therefore, we use the convexity to develop an $$O(M\log M)$$ O ( M log M ) algorithm that solves the problem on block graphs with M vertices.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:3:d:10.1007_s10957-015-0829-2
    DOI: 10.1007/s10957-015-0829-2
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    References listed on IDEAS

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    1. 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.
    2. Fahimeh Baroughi Bonab & Rainer Burkard & Elisabeth Gassner, 2011. "Inverse p-median problems with variable edge lengths," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 263-280, April.
    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. 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.
    5. Egon Balas & Eitan Zemel, 1980. "An Algorithm for Large Zero-One Knapsack Problems," Operations Research, INFORMS, vol. 28(5), pages 1130-1154, October.
    6. A. J. Goldman, 1971. "Optimal Center Location in Simple Networks," Transportation Science, INFORMS, vol. 5(2), pages 212-221, May.
    7. Kien Nguyen & André Chassein, 2015. "Inverse eccentric vertex problem 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. 23(3), pages 687-698, September.
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    Cited by:

    1. Trung Kien Nguyen & Nguyen Thanh Hung & Huong Nguyen-Thu, 2020. "A linear time algorithm for the p-maxian problem on trees with distance constraint," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1030-1043, November.
    2. 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.
    3. 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.
    4. Shahede Omidi & Jafar Fathali & Morteza Nazari, 2020. "Inverse and reverse balanced facility location problems with variable edge lengths on trees," OPSEARCH, Springer;Operational Research Society of India, vol. 57(2), pages 261-273, June.
    5. Alizadeh, Behrooz & Afrashteh, Esmaeil, 2020. "Budget-constrained inverse median facility location problem on tree networks," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    6. Ali Reza Sepasian, 2019. "Reverse 1-maxian problem with keeping existing 1-median," OPSEARCH, Springer;Operational Research Society of India, vol. 56(1), pages 1-13, March.
    7. Behrooz Alizadeh & Esmaeil Afrashteh & Fahimeh Baroughi, 2018. "Combinatorial Algorithms for Some Variants of Inverse Obnoxious Median Location Problem on Tree Networks," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 914-934, September.
    8. Nguyen, Kien Trung & Hung, Nguyen Thanh, 2021. "The minmax regret inverse maximum weight problem," Applied Mathematics and Computation, Elsevier, vol. 407(C).
    9. 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.
    10. Behrooz Alizadeh & Somayeh Bakhteh, 2017. "A modified firefly algorithm for general inverse p-median location problems under different distance norms," OPSEARCH, Springer;Operational Research Society of India, vol. 54(3), pages 618-636, September.
    11. 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.

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