Given a graph $G$ of $n$ nodes partitioned into facilities and customers, the $r$-edge interdiction covering problem (REIC) is to remove up to $r$ edges so as to maximize the total weight of customers disconnected from all facilities, which is called the covering objective function. While REIC is known to be NP-complete for general graphs, Fröhlich and Ruzika show that the problem can be solved in polynomial time when $G$ is a tree, providing an $O(n^7 r)$-time algorithm. We give an efficient $O(nr^2)$-time dynamic programming algorithm for REIC on trees that is fixed-parameter linear in $n$. Evaluating our solution on a benchmark of randomly generated tree networks with baselines of the Fröhlich and Ruzika algorithm and the Gurobi integer program solver, we demonstrate that in practice, our algorithm is both significantly faster and less sensitive to network topology and size. We extend our algorithm for REIC to graphs of bounded treewidth, a well-studied family of sparse graphs that generalizes trees, and obtain a matching runtime of $O(nr^2)$. We also consider the $r$-facility interdiction covering problem (RFIC), a novel variant of this network interdiction problem where the goal is to remove up to $r$ facilities to maximize the covering objective function over disconnected customers. We show that RFIC is NP-complete by observing it generalizes the small set bipartite vertex expansion problem (SSBVE), also known as the minimum $p$-union problem. We give an $O(nr^2)$-time algorithm for RFIC on trees, which also gives an $O(n^3)$-time algorithm for SSBVE on trees.

翻译：给定一个由 $n$ 个节点组成的图 $G$，节点划分为设施和客户两类，$r$-边阻断覆盖问题（REIC）的目标是移除至多 $r$ 条边，以最大化与所有设施断开连接的客户的总权重，该目标称为覆盖目标函数。虽然 REIC 在一般图上已知为 NP-完全问题，但 Fröhlich 和 Ruzika 表明当 $G$ 为树图时，该问题可在多项式时间内求解，并给出了一个 $O(n^7 r)$ 时间算法。我们针对树图上的 REIC 问题提出了一种高效的 $O(n r^2)$ 时间动态规划算法，该算法在 $n$ 上具有固定参数线性复杂度。通过在使用 Fröhlich 和 Ruzika 算法以及 Gurobi 整数规划求解器作为基线的随机生成树网络基准测试上评估我们的解，我们证明在实践中，我们的算法不仅显著更快，而且对网络拓扑和规模的敏感度更低。我们将 REIC 算法扩展到有界树宽图（一种通用树图的稀疏图族）上，并获得了匹配的 $O(n r^2)$ 运行时复杂度。我们还考虑了 $r$-设施阻断覆盖问题（RFIC），这是该网络阻断问题的一个新变体，其目标是通过移除至多 $r$ 个设施来最大化与断开连接客户相关的覆盖目标函数。我们通过观察 RFIC 推广了小集合二分顶点扩展问题（SSBVE，也称为最小 $p$-并问题），证明了 RFIC 是 NP-完全问题。我们给出了树图上 RFIC 问题的 $O(n r^2)$ 时间算法，该算法也同时给出了树图上 SSBVE 问题的 $O(n^3)$ 时间算法。