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The restricted inverse optimal value problem on shortest path under \(l_1\) norm on trees

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Abstract

We consider the restricted inverse optimal value problem on shortest path under weighted \(l_1\) norm on trees (RIOVSPT\(\varvec{_1}\)). It aims at adjusting some edge weights to minimize the total cost under weighted \(l_1\) 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 \(P_0\). If we ignore the restriction on the path \(P_0\), then we obtain the minimum cost shortest path interdiction problem on trees (MCSPIT\(\varvec{_1}\)). We analyze some properties of the problem (RIOVSPT\(\varvec{_1}\)) and explore the relationship of the optimal solutions between (MCSPIT\(\varvec{_1}\)) and (RIOVSPT\(\varvec{_1}\)). We first take the optimal solution of the problem (MCSPIT\(\varvec{_1}\)) as an initial infeasible solution of problem (RIOVSPT\(\varvec{_1}\)). Then we consider a slack problem \({\textbf {(}} {{\textbf {RIOVSPT}}}\varvec{_1^s)}\), where the length of the path \(P_0\) is greater than D. We obtain its feasible solutions gradually approaching to an optimal solution of the problem (RIOVSPT\(\varvec{_1}\)) by solving a series of subproblems \({{\textbf {(RIOVSPT}}}\varvec{_1^i)}\). It aims at determining the only weight-decreasing edge on the path \(P_0\) 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 \(P_0\) equals D. Consequently, the time complexity of the algorithm is \(O(n^2)\) and we present some numerical experiments to show the efficiency of the algorithm. Additionally, we devise a linear time algorithm for the problem (RIOVSPT\(\varvec{_{u1}}\)) under unit \(l_1\) norm.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Qiao Zhang, Xiucui Guan, Junhua Jia & Xinqiang Qian

  2. Center for Applied Optimization, University of Florida, Gainesville, FL, 32611, USA

    Panos M. Pardalos

  3. LATNA, Higher School of Economics, Nizhni Novgorod, Russia

    Panos M. Pardalos

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  1. Qiao Zhang
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  2. Xiucui Guan
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The work of X.C. Guan was supported by National Natural Science Foundation of China (11471073). The work of P.M. Pardalos was conducted within the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE).

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Zhang, Q., Guan, X., Jia, J. et al. The restricted inverse optimal value problem on shortest path under \(l_1\) norm on trees. J Glob Optim 86, 251–284 (2023). https://doi.org/10.1007/s10898-022-01256-7

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  • DOI: https://doi.org/10.1007/s10898-022-01256-7

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