Given an undirected graph with positive weights on the edges we study a parametric biobjective graph clustering problem. We remove a subset of edges to break the graph into smaller pieces, that is, connected components, or clusters. We seek to maximize the number of clusters while minimizing the weight of the removed edges. We identify nested solutions that lie on the concave envelope of the efficient frontier, yielding a hierarchical family of clusters, in strongly polynomial time. We demonstrate the performance of our approach on a graph defined by the schedule of football teams within the National Collegiate Athletic Association, which has a known hierarchical structure, and on a set of synthetic graphs generated from a stochastic block model with embedded hierarchical structure.

中文翻译：

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图上的最优层次聚类

给定一个边上权重为正的无向图，我们研究了一个参数双目标图聚类问题。我们删除边的子集以将图分解成更小的部分，即连接的组件或集群。我们寻求最大化集群的数量，同时最小化移除边缘的权重。我们在强多项式时间内确定了位于有效边界的凹包络上的嵌套解决方案，从而产生了一个层次化的集群族。我们在由具有已知层次结构的全国大学体育协会内的足球队时间表定义的图上以及从具有嵌入式层次结构的随机块模型生成的一组合成图上展示了我们的方法的性能。