We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.

翻译：我们考虑完全加权图中的一种聚类变体问题，目标是将节点划分为若干簇使得簇内边权重之和最大化。该问题被称为团划分问题，在边权重存在不同符号的一般情形下属于NP-hard问题。我们提出一种新的目标函数上界估计方法，并将其与经典分支定界技术相结合以求解精确解。我们在广泛范围的随机图和真实世界网络上评估了所提方法。与已知的替代方法相比，所提方法提供了更紧的上界，并实现了显著的收敛速度提升。