The Stochastic Critical Node Problem over Trees

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.