Network interdiction problems by upgading critical edges/nodes have important applications to reduce the infectivity of the COVID-19. A network of confirmed cases can be described as a rooted tree that has a weight of infectious intensity for each edge. Upgrading edges (nodes) can reduce the infectious intensity with contacts by taking prevention measures such as disinfection (treating the confirmed cases, isolating their close contacts or vaccinating the uninfected people). We take the sum of root-leaf distance on a rooted tree as the whole infectious intensity of the tree. Hence, we consider the sum of root-leaf distance interdiction problem by upgrading edges/nodes on trees (SDIPT-UE/N). The problem (SDIPT-UE) aims to minimize the sum of root-leaf distance by reducing the weights of some critical edges such that the upgrade cost under some measurement is upper-bounded by a given value. Different from the problem (SDIPT-UE), the problem (SDIPT-UN) aims to upgrade a set of critical nodes to reduce the weights of the edges adjacent to the nodes. The relevant minimum cost problem (MCSDIPT-UE/N) aims to minimize the upgrade cost on the premise that the sum of root-leaf distance is upper-bounded by a given value. We develop different norms to measure the upgrade cost. Under weighted Hamming distance, we show the problems (SDIPT-UE/N) and (MCSDIPT-UE/N) are NP-hard by showing the equivalence of the two problems and the 0–1 knapsack problem. Under weighted \(l_1\) norm, we solve the problems (SDIPT-UE) and (MCSDIPT-UE) in O(n) time by transforimg them into continuous knapsack problems. We propose two linear time greedy algorithms to solve the problem (SDIPT-UE) under unit Hamming distance and the problem (SDIPT-UN) with unit cost, respectively. Furthermore, for the the minimum cost problem (MCSDIPT-UE) under unit Hamming distance and the problem (MCSDIPT-UN) with unit cost, we provide two \(O(n\log n)\) time algorithms by the binary search methods. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.

通过更新关键边缘/节点来实现网络拦截问题对于降低 COVID-19 的传染性具有重要的应用价值。确诊病例网络可以描述为一棵有根树，每个边都有一个感染强度权重。升级边缘（节点）可以通过采取消毒等预防措施（治疗确诊病例、隔离密切接触者或为未感染者接种疫苗）来降低接触者的传染强度。我们将有根树上的根叶距离之和作为树的整体感染强度。因此，我们通过升级树上的边/节点（SDIPT-UE/N）来考虑根叶距离遮断总和问题。该问题（SDIPT-UE）旨在通过减少一些关键边的权重来最小化根叶距离的总和，从而使某些测量下的升级成本以给定值为上限。与问题（SDIPT-UE）不同，问题（SDIPT-UN）旨在升级一组关键节点以减少与节点相邻的边的权重。相关最小成本问题（MCSDIPT-UE/N）旨在在根叶距离之和为给定值的上限的前提下最小化升级成本。我们制定了不同的规范来衡量升级成本。在加权汉明距离下，我们通过展示这两个问题和 0-1 背包问题的等价性来证明问题 (SDIPT-UE/N) 和 (MCSDIPT-UE/N) 是 NP-hard。权重不足 该问题（SDIPT-UN）旨在升级一组关键节点以减少与节点相邻的边的权重。相关最小成本问题（MCSDIPT-UE/N）旨在在根叶距离之和为给定值的上限的前提下最小化升级成本。我们制定了不同的规范来衡量升级成本。在加权汉明距离下，我们通过展示这两个问题和 0-1 背包问题的等价性来证明问题 (SDIPT-UE/N) 和 (MCSDIPT-UE/N) 是 NP-hard。权重不足 该问题（SDIPT-UN）旨在升级一组关键节点以减少与节点相邻的边的权重。相关最小成本问题（MCSDIPT-UE/N）旨在在根叶距离之和为给定值的上限的前提下最小化升级成本。我们制定了不同的规范来衡量升级成本。在加权汉明距离下，我们通过展示这两个问题和 0-1 背包问题的等价性来证明问题 (SDIPT-UE/N) 和 (MCSDIPT-UE/N) 是 NP-hard。权重不足 我们制定了不同的规范来衡量升级成本。在加权汉明距离下，我们通过展示这两个问题和 0-1 背包问题的等价性来证明问题 (SDIPT-UE/N) 和 (MCSDIPT-UE/N) 是 NP-hard。权重不足 我们制定了不同的规范来衡量升级成本。在加权汉明距离下，我们通过展示这两个问题和 0-1 背包问题的等价性来证明问题 (SDIPT-UE/N) 和 (MCSDIPT-UE/N) 是 NP-hard。权重不足\(l_1\)范数，我们通过将问题 (SDIPT-UE) 和 (MCSDIPT-UE)转化为连续背包问题在O ( n ) 时间内解决它们。我们提出了两种线性时间贪心算法来分别解决单位汉明距离下的问题（SDIPT-UE）和单位成本问题（SDIPT-UN）。此外，对于单位汉明距离下的最小成本问题（MCSDIPT-UE）和单位成本问题（MCSDIPT-UN），我们通过二分搜索方法提供了两种\（O（n\log n）\）时间算法. 最后，我们进行了一些数值实验来比较这些算法获得的结果。