The Stochastic Critical Node Problem over Trees
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We tackle a stochastic version of the Critical Node Problem (CNP) where the goal is to minimize the pairwise connectivity of a graph by the deletion of a subset of its nodes. In the stochastic setting considered, the removal of nodes can fail with a certain probability. In our work we focus on tree graphs and demonstrate that over trees the stochastic CNP actually generalizes to the stochastic Critical Element Detection Problem where also the deletion of edges can fail with a certain probability. We also prove the NP-hardness of the problem while the deterministic counterpart was proved to be polynomial. We then derive linear and non-linear models for the considered CNP version. Moreover, we propose a Branch-and-Price approach for the problem and test its effectiveness on a large set of instances. As side result, we introduce an approximation algorithm for a problem variant of interest.
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