The maximum l-triangle k-club problem: Complexity, properties, and algorithms - Almeida, Maria Teresa; Brás, Raúl (Computers & operations research, 2019)
•The computational complexity of the problem is established, for any l ≥ 1 and k ≥ 2.•Cohesiveness properties of graphs induced by l-triangle k-clubs are derived.•New valid inequalities are presented.•Solution approaches based on alternative formulations of the problem are proposed.•Numerical results obtained on real-world networks are reported. Given a graph G = (V, E) and two positive integers l and k, an l-triangle k-club is a subset of nodes that induces a subgraph with each node included in at least l triplets linked pairwise and maximum distance between each pair of nodes at most k. This structure aims to represent cohesive groups in social and other complex networks. The Maximum l-Triangle k-Club Problem (MlTkCP) consists of finding a maximum cardinality l-triangle k-club of a given graph. In this paper, we derive properties of l-triangle k-clubs and show that the decision version of the MlTkCP is NP-Complete, for any given integers l ≥ 1 and k ≥ 2. To solve the problem, polynomial and non-polynomial formulations designed in different variable spaces are considered. The computational performance of exact solution approaches based on them is tested on a set of real-world graphs.