# Casting light on the hidden bilevel combinatorial structure of the capacitated vertex separator problem

Given an undirected graph, we study the capacitated vertex separator problem that asks to find a subset of vertices of minimum cardinality, the removal of which induces a graph having a bounded number of pairwise disconnected shores (subsets of vertices) of limited cardinality. The problem is of great importance in the analysis and protection of communication or social networks against possible viral attacks and for matrix decomposition algorithms. In this article, we provide a new bilevel interpretation of the problem and model it as a two-player Stackelberg game in which the leader interdicts the vertices (i.e., decides on the subset of vertices to remove), and the follower solves a combinatorial optimization problem on the resulting graph. This approach allows us to develop a computational framework based on an integer programming formulation in the natural space of the variables. Link to the article

FURINI, F., LJUBIC, I., MALAGUTI, E. and PARONUZZI, P. (2022). Casting light on the hidden bilevel combinatorial structure of the capacitated vertex separator problem. *Operations Research*, 70(4), pp. 2399–2420.

Keywords : #Networks/graphs, #Integer-programming, #Benders-decomposition, #Cutting-plane