Minimum-Cost Shortest-Path Interdiction Problem Involving Upgrading Edges on Trees with Weighted l_∞ Norm

Network interdiction problems involving edge deletion on shortest paths have wide applications. However, in many practical scenarios, the complete removal of edges is infeasible. The minimum-cost shortest-path interdiction problem for trees with the weighted $l_{\infty }$ norm (MCSPIT${}_{\infty }$) is studied in this paper. The goal is to upgrade selected edges at minimum total cost such that the shortest root–leaf distance is bounded below by a given value. We designed an $O(nlogn)$ algorithm based on greedy techniques combined with a binary search method to solve this problem efficiently. We then extended the framework to the minimum-cost shortest-path double interdiction problem for trees with the weighted $l_{\infty }$ norm, which imposes an additional requirement that the sum of root–leaf distances exceed a given threshold. Building upon the solution to (MCSPIT${}_{\infty }$), we developed an equally efficient $O(nlogn)$ algorithm for this variant. Finally, numerical experiments are presented to demonstrate both the effectiveness and practical performance of the proposed algorithms.