Efficient Detection of $k$-Plex Structures in Large Graphs Through Constraint Learning

The $k$-plex is a popular definition of communities in networks, offering more flexibility than cliques by allowing each node to miss up to $k$ connections. However, finding $k$-plexes in large graphs is a theoretically challenging task due to the large number of possible $k$-plexes. In this article, we propose a novel approach for detecting $k$-plexes under various sizes and time constraints using an automated strategy to learn bounds, called the constraint learning and bounding (CLB) method. Specifically, our proposed CLB approach, leverages the concept of constraint learning to develop a mixed integer linear programming (MILP) instance as a model to learn a bounding strategy in the branch-and-bound process. The variables in the MILP instances correspond to the natural properties of the $k$-plex problem. Unlike previous works, we focus on learning the bounding strategy rather than learning the branching strategy. Thus, the strategy learned by our proposed approach avoids visiting infeasible solutions, which accelerates the branch-and-bound algorithm and reduces the computational load. To evaluate our approach, we conduct experiments on various real graphs to validate the superiority and the generality of our proposed approach. In summary, our approach offers an effective and efficient solution for detecting $k$-plexes under various conditions.