No abstract

Given a graph with its vertex set partitioned into a set of groups, nonnegative costs associated to its edges, and nonnegative prizes associated to its vertices, the prize‐collecting generalized minimum spanning tree problem consists in finding a subtree of this graph that spans exactly one vertex of each group and minimizes the sum of the costs of the edges of the tree less the prizes of the selected vertices. It is a generalization of the NP‐hard generalized minimum spanning tree optimization problem. We propose a GRASP (greedy randomized adaptive search procedure) heuristic for its approximate solution, incorporating path‐relinking for search intensification and a restart strategy for search diversification. The hybridization of the GRASP with path‐relinking and restarts heuristic with a data mining strategy that is applied along with the GRASP iterations, after the elite set is modified and becomes stable, contributes to making the heuristic more robust. The computational experiments show that the heuristic developed in this work found very good solutions for test problems with up to 439 vertices. All input data for the test instances and detailed numerical results are made available from Mendeley Data.

The longest induced path problem consists in finding a maximum subset of vertices of a graph such that it induces a simple path. We propose a new exact enumerative algorithm that solves problems with up to 138 vertices and 493 edges and a heuristic for larger problems. Detailed computational experiments compare the results obtained by the new algorithms with other approaches in the literature and investigate the characteristics of the optimal solutions.

Polarization arises when the underlying network connecting the members of a community or society becomes characterized by highly connected groups with weak intergroup connectivity. The increasing polarization, the strengthening of echo chambers, and the isolation caused by information filters in social networks are increasingly attracting the attention of researchers from different areas of knowledge such as computer science, economics, and social and political sciences. This work presents an annotated review of network polarization measures and models used to handle the polarization. Several approaches for measuring polarization in graphs and networks were identified, including those based on homophily, modularity, random walks, and balance theory. The strategies used for reducing polarization include methods that propose edge or node editions (including insertions or deletions as well as edge weight modifications), changes in social network design, or changes in the recommendation systems embedded in these networks.