Combinatorial algorithms for restricted inverse optimal value problems on minimum spanning tree under weighted l1 norm - Jia, Junhua; Guan, Xiucui; Zhang, Binwu; Qian, Xinqiang; Pardalos, Panos M. (Journal of computational and applied mathematics, 2024)
We study the restricted inverse optimal value problem on minimum spanning tree under weighted l1 norm. In a connected edge-weighted network G(V,E,w), we are given a spanning tree T0, a cost vector c and a value K. We aim to obtain a new weight vector w̄ satisfying the bounded constraint such that T0 is a minimum spanning tree under w̄ whose weight is just K. We focus on minimizing the modification cost under weighted l1 norm. We first convert its mathematical model into a linear programming problem (P). Then we solve its dual problem (D) by a sub-problem (Dz∗) corresponding to the critical value z∗ which can be calculated by a binary search method. Impressively the sub-problem Dz for z∈R can be transformed into a minimum cost flow problem on an auxiliary network. Finally, we propose an O(|E|2|V|2log|V|log(|V|cmax)) algorithm, where cmax is the maximum cost of the vector c.