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：我们为完整的加权图表考虑组别问题的变体。 目的是将节点分成组, 使组群中边缘重量的总和最大化。 这个问题被称为分层分割问题, 在有不同标志的边缘重量的一般情况下, 这个问题是NP硬的。 我们提出了一种新的方法来估计目标功能的上层界限, 即我们与传统分支和约束技术相结合, 以找到确切的解决办法。 我们评估了我们对于范围广泛的随机图和真实世界网络的做法。 与已知的替代方法相比, 拟议的方法提供了更严格的上层界限, 并实现了显著的趋同速度改进。