We tackle a stochastic version of the Critical Node Problem (CNP) where the goal is to minimize the pairwise connectivity of a graph by attacking a subset of its nodes. In the stochastic setting considered, the attacks on nodes can fail with a certain probability. In our work we focus on trees and demonstrate that over trees the stochastic CNP actually generalizes to the stochastic Critical Element Detection Problem where attacks on edges can also fail with a certain probability. We also prove the NP-completeness of the decision version of the problem when connection costs are one, while its deterministic counterpart was proved to be polynomial. We then derive linear and nonlinear models for the considered CNP version. Moreover, we propose an exact approach based on Benders decomposition and test its effectiveness on a large set of instances. As a side result, we introduce an approximation algorithm for a problem variant of interest.

中文翻译：

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树上的随机临界节点问题

我们解决了关键节点问题 (CNP) 的随机版本，其目标是通过攻击其节点的子集来最小化图的成对连通性。在考虑的随机设置中，对节点的攻击可能以一定的概率失败。在我们的工作中，我们专注于树并证明在树上随机 CNP 实际上推广到随机关键元素检测问题，其中对边缘的攻击也可能以一定的概率失败。我们还证明了当连接成本为 1 时问题决策版本的 NP 完全性，而其确定性对应项被证明是多项式。然后，我们为所考虑的 CNP 版本推导出线性和非线性模型。此外，我们提出了一种基于 Benders 分解的精确方法，并在大量实例上测试其有效性。