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

We consider the restricted inverse optimal value problem on shortest path under weighted l_1 norm on trees (RIOVSPT _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 _1 ) . We analyze some properties of the problem (RIOVSPT _1 ) and explore the relationship of the optimal solutions between (MCSPIT _1 ) and (RIOVSPT _1 ) . We first take the optimal solution of the problem (MCSPIT _1 ) as an initial infeasible solution of problem (RIOVSPT _1 ) . Then we consider a slack problem (RIOVSPT_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 _1 ) by solving a series of subproblems (RIOVSPT_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 _u1 ) under unit l_1 norm.