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Maximum shortest path interdiction problem by upgrading edges on trees under weighted $$l_1$$ l 1 norm

Author

Listed:
  • Qiao Zhang

    (Southeast University)

  • Xiucui Guan

    (Southeast University)

  • Panos M. Pardalos

    (University of Florida
    LATNA, Higher School of Economics)

Abstract

Network interdiction problems by deleting critical edges have wide applicatio ns. However, in some practical applications, the goal of deleting edges is difficult to achieve. We consider the maximum shortest path interdiction problem by upgrading edges on trees (MSPIT) under unit/weighted $$l_1$$ l 1 norm. We aim to maximize the the length of the shortest path from the root to all the leaves by increasing the weights of some edges such that the upgrade cost under unit/weighted $$l_1$$ l 1 norm is upper-bounded by a given value. We construct their mathematical models and prove some properties. We propose a revised algorithm for the problem (MSPIT) under unit $$l_1$$ l 1 norm with time complexity O(n), where n is the number of vertices in the tree. We put forward a primal dual algorithm in $$O(n^2)$$ O ( n 2 ) time to solve the problem (MSPIT) under weighted $$l_1$$ l 1 norm, in which a minimum cost cut is found in each iteration. We also solve the problem to minimize the cost to upgrade edges such that the length of the shortest path is lower bounded by a value and present an $$O(n^2)$$ O ( n 2 ) algorithm. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.

Suggested Citation

  • Qiao Zhang & Xiucui Guan & Panos M. Pardalos, 2021. "Maximum shortest path interdiction problem by upgrading edges on trees under weighted $$l_1$$ l 1 norm," Journal of Global Optimization, Springer, vol. 79(4), pages 959-987, April.
    • Handle: RePEc:spr:jglopt:v:79:y:2021:i:4:d:10.1007_s10898-020-00958-0
    DOI: 10.1007/s10898-020-00958-0
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    References listed on IDEAS

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    1. Cristina Bazgan & Sonia Toubaline & Daniel Vanderpooten, 2013. "Complexity of determining the most vital elements for the p-median and p-center location problems," Journal of Combinatorial Optimization, Springer, vol. 25(2), pages 191-207, February.
    2. Huili Zhang & Yinfeng Xu & Xingang Wen, 2015. "Optimal shortest path set problem in undirected graphs," Journal of Combinatorial Optimization, Springer, vol. 29(3), pages 511-530, April.
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    Cited by:

    1. 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.
    2. Qiao Zhang & Xiucui Guan & Junhua Jia & Xinqiang Qian & Panos M. Pardalos, 2023. "The restricted inverse optimal value problem on shortest path under $$l_1$$ l 1 norm on trees," Journal of Global Optimization, Springer, vol. 86(1), pages 251-284, May.

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