The quickest root–leaf interdiction problem on tree networks

We investigate the quickest root–leaf interdiction problem on a tree, in which two players compete to achieve their objectives. The evader aims to escape the tree by traversing from the root to one of the leaves with the minimum transmission time, while the interdictor seeks to reduce the edge capacities within a given budget to maximize the transmission time of the evader in the perturbed tree. We prove that if the modifying cost is measured using the Hamming distance and the adjusted capacities are limited to a threshold, then there is an optimal solution, where the modified capacities are either equal to themselves or equal to the threshold. We also demonstrate that there are a linearly large number of values that can serve as candidates for the optimal objective. For each fixed value, we construct an auxiliary network in which its minimum cut is associated with the given value. The minimum cut approach helps to develop a combinatorial algorithm that solves the corresponding problem in $O(nmax\{m,logn\})$ time, where $n$ and $m$ are respectively the number of vertices and leaves in the underlying tree.