A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence, we consider the SRD interdiction problem on trees with cardinality constraint by upgrading edges, denoted by (SDIPTC). It aims to maximize the SRD by upgrading the weights of N critical edges such that the total upgrade cost under some measurement is upper-bounded by a given value. The relevant minimum cost problem (MCSDIPTC) aims to minimize the total upgrade cost on the premise that the SRD is lower-bounded by a given value. We develop two different norms including weighted \(l_\infty \) norm and weighted bottleneck Hamming distance to measure the upgrade cost. We propose two binary search algorithms within O(\(n\log n\)) time for the problems (SDIPTC) under the two norms, respectively. For problems (MCSDIPTC), we propose two binary search algorithms within O(\(N n^2\)) and O(\(n \log n\)) under weighted \(l_\infty \) norm and weighted bottleneck Hamming distance, respectively. These problems are solved through their subproblems (SDIPT) and (MCSDIPT), in which we ignore the cardinality constraint on the number of upgraded edges. Finally, we design numerical experiments to show the effectiveness of these algorithms.

用于物资运输的网络可以描述为一棵有根的树，每条边的拥塞权重为一定程度。我们将有根树上的根-叶距离 （SRD） 之和作为树的整个拥塞程度。因此，我们通过升级边缘来考虑具有基数约束的树上的 SRD 拦截问题，用 （SDIPTC） 表示。它旨在通过升级 N 个关键边缘的权重来最大化 SRD，以便在某些测量下的总升级成本与给定值的上限相同。相关的最低成本问题 （MCSDIPTC） 旨在在 SRD 的下限为给定值的前提下最小化总升级成本。我们开发了两种不同的范数，包括加权 \（l_\infty \） 范数和加权瓶颈汉明距离来衡量升级成本。我们分别针对两个范数下的问题 （SDIPTC） 在 O（\（n\log n\）） 时间内提出了两种二叉搜索算法。对于问题 （MCSDIPTC），我们提出了两种二叉搜索算法，分别在加权 \（l_\infty \） 范数和加权瓶颈汉明距离下，分别为 O（\（N n^2\）） 和 O（\（n \log n\）） 两种。这些问题通过它们的子问题 （SDIPT） 和 （MCSDIPT） 来解决，在这些子问题中，我们忽略了对升级边数量的基数约束。最后，我们设计了数值实验来证明这些算法的有效性。