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arXiv 2407.14906cs.DScs.DM
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最小生成树及其他拟阵基的阻断问题

Interdiction of minimum spanning trees and other matroid bases

Noah Weninger, Ricardo Fukasawa

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29 pages, 2 figures
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AI总结：

本文提出一种基于动态规划上界和分支定界的精确算法，用于求解拟阵阻断问题，在划分拟阵上达到伪多项式时间，在图拟阵上通过最小割近似，显著加速了最小生成树阻断问题的求解。

摘要：

在最小生成树（MST）阻断问题中，给定一个带边权的图$G=(V,E)$，我们希望找到满足背包约束的某个$X\subseteq E$，使得在$(V,E\setminus X)$中MST的权重最大化。由于$G$的MST是$G$的图拟阵中的最小权重基，该问题是拟阵$M=(E,\mathcal{I})$上拟阵阻断问题的特例，其目标改为最大化与$X$不相交的$M$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。我们开发了一种新的精确算法来解决拟阵阻断问题。我们算法的关键组成部分之一是一个动态规划上界，该上界仅要求对于给定的拟阵，可以计算/近似一个更简单的离散导数问题。然后我们的精确算法在自定义的分支定界算法中使用这个界。对于不同的拟阵，我们展示了如何计算/近似这个离散导数。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，可以通过求解一系列最小割问题来获得近似，我们将其应用于MST阻断问题。我们算法的运行时间在渐近意义上比已知最佳的MST阻断算法更快，直到多对数因子。此外，我们的算法实现了最先进的计算性能：我们解决了文献中所有可用的实例，并在许多情况下将最佳运行时间从几小时减少到几秒。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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