Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem

Exploiting Bilevel Optimization Techniques to Disconnect Graphs into Small Components In order to limit the spread of possible viral attacks in a communication or social network, it is necessary to identify critical nodes, the protection of which disconnects the remaining unprotected graph into a bounded number of shores (subsets of vertices) of limited cardinality. In the article “'Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem”, Furini, Ljubic, Malaguti, and Paronuzzi provide a new bilevel interpretation of the associated capacitated vertex separator problem and model it as a two-player Stackelberg game in which the leader interdicts (protects) the vertices, and the follower solves a combinatorial optimization problem on the resulting graph. Thanks to this bilevel interpretation, the authors derive different families of strengthening inequalities and show that they can be separated in polynomial time. The ideas exploited in their framework can also be extended to other vertex/edge deletion/insertion problems or graph partitioning problems by modeling them as two-player Stackelberg games to be solved through bilevel optimization.

Inverse Percolation to Quantify Robustness in Multiplex Networks

Inverse percolation is known as the problem of finding the minimum set of nodes whose elimination of their links causes the rupture of the network. Inverse percolation has been widely used in various studies of single-layer networks. However, the use and generalization of multiplex networks have been little considered. In this work, we propose a methodology based on inverse percolation to quantify the robustness of multiplex networks. Specifically, we present a modified version of the mathematical model for the multiplex-vertex separator problem (m-VSP). By solving the m-VSP, we can find nodes that cause the rupture of the mutually connected giant component (MCGC) and the large viable cluster (LVC) when their links are removed from the network. The methodology presented in this work was tested in a set of benchmark networks, and as case study, we present an analysis using a set of multiplex social networks modeled with information about the main characteristics of the best universities in the world and the universities in Mexico. The results show that the methodology presented in this work can work in different models and types of 2- and 3-layer multiplex networks without dividing the entire multiplex network into single-layer as some techniques described in the specific literature. Furthermore, thanks to the fact that the technique does not require the calculation of some structural measure or centrality metric, and it is easy to scale for networks of different sizes.