Combinatorial algorithms for restricted inverse optimal value problems on minimum spanning tree under weighted l1 norm

• Published in: Journal of computational and applied mathematics Vol. 451
• Main Authors: Jia, Junhua, Guan, Xiucui, Zhang, Binwu, Qian, Xinqiang, Pardalos, Panos M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2024
• Subjects: Dual theory; l1 norm; Linear programming; Minimum cost flow; Minimum spanning tree; Restricted inverse optimal value problem; Restricted inverse optimal value problem; Minimum cost flow; Linear programming; l1 norm; Dual theory; Minimum spanning tree
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2024.116110

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.