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

Qiao Zhang, Xiucui Guan (), Junhua Jia, Xinqiang Qian and Panos M. Pardalos
Additional contact information
Qiao Zhang: Southeast University
Xiucui Guan: Southeast University
Junhua Jia: Southeast University
Xinqiang Qian: Southeast University
Panos M. Pardalos: University of Florida

Journal of Global Optimization, 2023, vol. 86, issue 1, No 10, 284 pages

Abstract: 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.

Keywords: Inverse optimal value problem; Shortest path; Tree; $$l_1$$ l 1 norm; Minimum cost cut (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10898-022-01256-7

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