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The inverse 1-center problem on trees with variable edge lengths under Chebyshev norm and Hamming distance

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  • Kien Trung Nguyen

    (Cantho University)

  • Ali Reza Sepasian

    (Fasa University)

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 $$O(n\log n)$$ O ( n log n ) time, where $$n+1$$ n + 1 is the number of vertices of the tree. Dropping this condition, the problem can be solved in $$O(n^{2})$$ O ( n 2 ) time.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:32:y:2016:i:3:d:10.1007_s10878-015-9907-5
    DOI: 10.1007/s10878-015-9907-5
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Xiucui Guan & Binwu Zhang, 2012. "Inverse 1-median problem on trees under weighted Hamming distance," Journal of Global Optimization, Springer, vol. 54(1), pages 75-82, September.
    4. 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.
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    Cited by:

    1. Roghayeh Etemad & Behrooz Alizadeh, 2018. "Reverse selective obnoxious center location problems on tree graphs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 431-450, June.
    2. Alizadeh, Behrooz & Afrashteh, Esmaeil, 2020. "Budget-constrained inverse median facility location problem on tree networks," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    3. 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.
    4. Xianyue Li & Xichao Shu & Huijing Huang & Jingjing Bai, 2019. "Capacitated partial inverse maximum spanning tree under the weighted Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 38(4), pages 1005-1018, November.
    5. 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.
    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. 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.
    9. 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.
    10. 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.
    11. 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.

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