The family of critical node detection problems asks for finding a subset of vertices, deletion of which minimizes or maximizes a predefined connectivity measure on the remaining network. We study a problem of this family called the k-vertex cut problem. The problem asks for determining the minimum weight subset of nodes whose removal disconnects a graph into at least k components. We provide two new integer linear programming formulations, along with families of strengthening valid inequalities. Both models involve an exponential number of constraints for which we provide poly-time separation procedures and design the respective branch-and-cut algorithms. In the first formulation one representative vertex is chosen for each of the k mutually disconnected vertex subsets of the remaining graph. In the second formulation, the model is derived from the perspective of a two-phase Stackelberg game in which a leader deletes the vertices in the first phase, and in the second phase a follower builds connected components in the remaining graph. Our computational study demonstrates that a hybrid model in which valid inequalities of both formulations are combined significantly outperforms the state-of-the-art exact methods from the literature.

## Dettaglio pubblicazione

2020, MATHEMATICAL PROGRAMMING COMPUTATION, Pages 133-164 (volume: 12)

### On integer and bilevel formulations for the k-vertex cut problem (*01a Articolo in rivista*)

#### Furini F., Ljubic I., Malaguti E., Paronuzzi P.

**Gruppo di ricerca:**Combinatorial Optimization

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