In the minimum spanning tree (MST) interdiction problem, we are given a graph $G=(V,E)$ with edge weights, and want to find some $X\subseteq E$ satisfying a knapsack constraint such that the MST weight in $(V,E\setminus X)$ is maximized. Since MSTs of $G$ are the minimum weight bases in the graphic matroid of $G$, this problem is a special case of matroid interdiction on a matroid $M=(E,\mathcal{I})$, in which the objective is instead to maximize the minimum weight of a basis of $M$ which is disjoint from $X$. By reduction from 0-1 knapsack, matroid interdiction is NP-complete, even for uniform matroids. We develop a new exact algorithm to solve the matroid interdiction problem. One of the key components of our algorithm is a dynamic programming upper bound which only requires that a simpler discrete derivative problem can be calculated/approximated for the given matroid. Our exact algorithm then uses this bound within a custom branch-and-bound algorithm. For different matroids, we show how this discrete derivative can be calculated/approximated. In particular, for partition matroids, this yields a pseudopolynomial time algorithm. For graphic matroids, an approximation can be obtained by solving a sequence of minimum cut problems, which we apply to the MST interdiction problem. The running time of our algorithm is asymptotically faster than the best known MST interdiction algorithm, up to polylog factors. Furthermore, our algorithm achieves state-of-the-art computational performance: we solved all available instances from the literature, and in many cases reduced the best running time from hours to seconds.

翻译：在最小生成树（MST）阻断问题中，给定一个带边权重的图$G=(V,E)$，目标是找到满足背包约束的边集$X\subseteq E$，使得在$(V,E\setminus X)$中的最小生成树权重最大化。由于$G$的最小生成树是其图拟阵中的最小权基，该问题是拟阵阻断问题的一个特例：在拟阵$M=(E,\mathcal{I})$中，目标转化为最大化与$X$不交的$M$的基的最小权重。通过从0-1背包问题归约，即使对于均匀拟阵，拟阵阻断问题也是NP完全的。本文提出了一种求解拟阵阻断问题的新精确算法。算法的关键组成部分之一是一个动态规划上界，该上界仅要求对给定拟阵能计算/近似一个更简单的离散导数问题。我们的精确算法在定制的分支定界框架中利用此上界。针对不同拟阵，我们展示了如何计算/近似该离散导数。特别地，对于划分拟阵，这产生了一个伪多项式时间算法。对于图拟阵，可通过求解一系列最小割问题来获得近似解，我们将此应用于MST阻断问题。本算法的渐进时间复杂度优于已知最优的MST阻断算法（相差多对数因子）。此外，算法取得了当前最佳的计算性能：我们成功求解了文献中所有可用算例，并在多数情况下将最优运行时间从数小时缩短至数秒。