Abstract
Abstract The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k , the maximum edge subgraph problem consists in finding a k -vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families.
Highlights
Social network analysis (SNA) is an important tool to study the relationships and flows between people, organizations, and other entities
We study the polytope associated to the formulation MIP1 of maximum edge subgraph problem (MESP) introduced by [3], introduce several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families
In this work we have presented a polyhedral study of the maximum edge subgraph problem, by introducing seven families of valid inequalities
Summary
Social network analysis (SNA) is an important tool to study the relationships and flows between people, organizations, and other entities. The detection of quasi-cliques is crucial in [7] for studying the network of bilateral investment treaties. In this case, quasi-cliques are used both in the analysis of cohesive subgroups and as an instrument to evaluate differences in the topology of random graphs. There are two main approaches to study quasi-cliques: (a) given a specified edge density γ ∈ [0, 1], find the largest vertex set which is γ-dense and, (b) given a size k, find the densest set of k vertices. The second approach is known in the graph and optimization literature as the maximum edge subgraph problem (MESP) or dense/densest/heaviest k-subgraph problem. We study the polytope associated to the formulation MIP1 of MESP introduced by [3], introduce several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families
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