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

Journal of Global Optimization. 2023. Vol. 86. P. 251–284.

Zhang Q., Guan X., Jia J., Qian X., Pardalos P. M.

We consider the restricted inverse optimal value problem on shortest path under weighted l1 norm on trees (RIOVSPT1). It aims at adjusting some edge weights to minimize the total cost under weighted l1 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 P0. If we ignore the restriction on the path P0, then we obtain the
minimum cost shortest path interdiction problem on trees (MCSPIT1). We analyze some properties of the problem (RIOVSPT1) and explore the relationship of the optimal solutions between (MCSPIT1) and (RIOVSPT1). We first take the optimal solution of the problem (MCSPIT1) as an initial infeasible solution of problem (RIOVSPT1). Then we consider a slack problem (RIOVSPTs1 ), where the length of the path P0 is greater than D. We obtain its
feasible solutions gradually approaching to an optimal solution of the problem (RIOVSPT1) by solving a series of subproblems (RIOVSPTi1). It aims at determining the only weightdecreasing edge on the path P0 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 P0 equals D. Consequently, the time complexity of the algorithm is O(n2) and we present some numerical experiments to show the efficiency of the algorithm. Additionally, we devise a linear time algorithm for the problem (RIOVSPTu1) under unit l1 norm.

Keywords: Shortest path, Minimum cost cut, Tree, Inverse optimal value problem, l1 norm

Publication based on the results of: Research on graph and network structures and its applications (2023)