Given a graph G=(V,E) and a threshold γ∈(0,1], the maximum quasi‐clique problem amounts to finding a maximum cardinality subset C∗ of the vertices in V such that the edge density of the graph induced in G by C∗ is greater than or equal to the threshold. This problem is NP‐hard and has a number of applications in data mining, for example, in social networks or phone call graphs. In this work, we present an exact algorithm to solve this problem, based on a quasi‐hereditary property. We also propose a new upper bound that is used for pruning the search tree. Numerical results show that the new approach is competitive and outperforms the best integer programming approaches in the literature. The new upper bound is consistently tighter than previously existing bounds.

Given a graph $G=(V,E)$ and a threshold $\gamma \in (0,1]$, the maximum cardinality quasi-clique problem consists in finding a maximum cardinality subset $C^*$ of the vertices in $V$ such that the density of the graph induced in $G$ by $C^*$ is greater than or equal to the threshold $\gamma$. This problem has a number of applications in data mining, e.g. in social networks or phone call graphs. We propose a matheuristic for solving the maximum cardinality quasi-clique problem, based on the hybridization of a biased random-key genetic algorithm (BRKGA) with an exact local search strategy. The newly proposed approach is compared with a pure biased random-key genetic algorithm, which was the best heuristic in the literature at the time of writing. Computational results show that the hybrid BRKGA outperforms the pure BRKGA.

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