In the Vertex Triangle 2-Club problem, we are given an undirected graph G and aim to find a maximum-vertex subgraph of G that has diameter at most 2 and in which every vertex is contained in at least \(\ell \) triangles in the subgraph. So far, the only algorithm for solving Vertex Triangle 2-Club relies on an ILP formulation (Almeida and Brás in Comput Oper Res 111:258–270, 2019). In this work, we develop a combinatorial branch-and-bound algorithm that, coupled with a set of data reduction rules, outperforms the existing implementation and is able to find optimal solutions on sparse real-world graphs with more than 100,000 vertices in a few minutes. We also extend our algorithm to the Edge Triangle 2-Club problem where the triangle constraint is imposed on all edges of the subgraph.

在顶点三角形 2-Club问题中，给定一个无向图G ，目标是找到G的最大顶点子图，其直径至多为 2，并且其中每个顶点至少包含在\(\ell \)个三角形中在子图中。到目前为止，求解Vertex Triangle 2-Club的唯一算法依赖于 ILP 公式（Almeida 和 Brás in Comput Oper Res 111:258–270, 2019）。在这项工作中，我们开发了一种组合分支定界算法，与一组数据缩减规则相结合，其性能优于现有实现，并且能够在具有超过 100,000 个顶点的稀疏现实世界图上找到最优解。分钟。我们还将我们的算法扩展到边三角形 2-Club问题，其中三角形约束施加在子图的所有边上。