论文标题
未加权图的分层聚类
Hierarchical Clusterings of Unweighted Graphs
论文作者
论文摘要
我们研究了在最近引入的Dasgupta目标函数下找到未加权相似图的最佳分层聚类的复杂性。我们介绍了一种称为标准化过程的证明技术，该技术采用了图$ g $的任何此类聚类，并迭代地对其进行了改进，直到达到G达到G的目标目标聚类为止。我们使用此技术显示负面的复杂性和积极的复杂性。首先，我们通常表明问题是NP完整的。 Secondly, we consider min-well-behaved graphs, which are graphs $H$ having the property that for any $k$ the graph $H(k)$ being the join of $k$ copies of $H$ has an optimal hierarchical clustering that splits each copy of $H$ in the same optimal way.为了最佳地群集这样的图形$ h（k）$，因此我们只需要最佳地聚类较小的图形$ h $。联合两分的图是最小行为的，但否则它们似乎很少。我们使用归一化过程来表明6个顶点的周期也是最小行为。
We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph $G$ and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs $H$ having the property that for any $k$ the graph $H(k)$ being the join of $k$ copies of $H$ has an optimal hierarchical clustering that splits each copy of $H$ in the same optimal way. To optimally cluster such a graph $H(k)$ we thus only need to optimally cluster the smaller graph $H$. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.
