Computing maximum k-defective cliques in massive graphs

•Properties of the maximum k-defective clique problem.•A novel branch-and-bound based exact algorithm.•Graph preprocessing, two-hop reduction and color-bound techniques for improving the performance of branch-and-bound.•Numerical results on a wide range of real graphs. A k-defective clique (k is a non-negative integer) of an undirected graph G is a subset of its vertices, which induces a nearly complete graph with a maximum of k missing edges. The maximum k-defective clique problem (MDCP) is to determine the k-defective clique of the maximum size in the graph. As a relaxation of the popular maximum clique problem, the MDCP is a relevant model for a number of practical applications such as complex network analysis. However, it is computationally challenging to solve the problem. In this study, we investigate a set of general and dedicated graph reduction and pruning techniques to improve exact search algorithms based on the branch-and-bound framework. We present results of extensive computational experiments on 141 benchmark graphs from several popular sources, including both random graphs and massive real-world networks. Comparisons with two state-of-the-art methods in the literature demonstrate that our approach is on par with the reference methods and performs remarkably well on massive graphs.