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The restricted inverse optimal value problem on shortest path under $$l_1$$ l 1 norm on trees

Author

Listed:
  • Qiao Zhang

    (Southeast University)

  • Xiucui Guan

    (Southeast University)

  • Junhua Jia

    (Southeast University)

  • Xinqiang Qian

    (Southeast University)

  • Panos M. Pardalos

    (University of Florida
    LATNA, Higher School of Economics)

Abstract

We consider the restricted inverse optimal value problem on shortest path under weighted $$l_1$$ l 1 norm on trees (RIOVSPT $$\varvec{_1}$$ 1 ). It aims at adjusting some edge weights to minimize the total cost under weighted $$l_1$$ l 1 norm on the premise that the length of the shortest root-leaf path of the tree is lower-bounded by a given value D, which is just the restriction on the length of a given root-leaf path $$P_0$$ P 0 . If we ignore the restriction on the path $$P_0$$ P 0 , then we obtain the minimum cost shortest path interdiction problem on trees (MCSPIT $$\varvec{_1}$$ 1 ). We analyze some properties of the problem (RIOVSPT $$\varvec{_1}$$ 1 ) and explore the relationship of the optimal solutions between (MCSPIT $$\varvec{_1}$$ 1 ) and (RIOVSPT $$\varvec{_1}$$ 1 ). We first take the optimal solution of the problem (MCSPIT $$\varvec{_1}$$ 1 ) as an initial infeasible solution of problem (RIOVSPT $$\varvec{_1}$$ 1 ). Then we consider a slack problem $${\textbf {(}} {{\textbf {RIOVSPT}}}\varvec{_1^s)}$$ ( RIOVSPT 1 s ) , where the length of the path $$P_0$$ P 0 is greater than D. We obtain its feasible solutions gradually approaching to an optimal solution of the problem (RIOVSPT $$\varvec{_1}$$ 1 ) by solving a series of subproblems $${{\textbf {(RIOVSPT}}}\varvec{_1^i)}$$ ( RIOVSPT 1 i ) . It aims at determining the only weight-decreasing edge on the path $$P_0$$ P 0 with the minimum cost so that the length of the shortest root-leaf path is no less than D. The subproblem can be solved by searching for a minimum cost cut in O(n) time. The iterations continue until the length of the path $$P_0$$ P 0 equals D. Consequently, the time complexity of the algorithm is $$O(n^2)$$ O ( n 2 ) and we present some numerical experiments to show the efficiency of the algorithm. Additionally, we devise a linear time algorithm for the problem (RIOVSPT $$\varvec{_{u1}}$$ u 1 ) under unit $$l_1$$ l 1 norm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:1:d:10.1007_s10898-022-01256-7
    DOI: 10.1007/s10898-022-01256-7
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    References listed on IDEAS

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    1. Hui Wang & Xiucui Guan & Qiao Zhang & Binwu Zhang, 2021. "Capacitated inverse optimal value problem on minimum spanning tree under bottleneck Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 861-887, May.
    2. 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.
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