Abstract

We consider the computational complexity of one extremal problem of choosing a subset of \(p \) points from some given \(2 \)-clustering of a finite set in a metric space. The chosen subset of points has to describe the given clusters in the best way from the viewpoint of some geometric criterion. This is a formalization of an applied problem of data mining which consists in finding a subset of typical representatives of a dataset composed of two classes based on the function of rival similarity. The problem is proved to be NP-hard. To this end, we polynomially reduce to the problem one of the well-known problems NP-hard in the strong sense, the \(p \)-median problem.

中文翻译：

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在$$ \ boldsymbol 2 $$中选择典型代表的问题的计算复杂性-公制空间中一组有限点的聚类

摘要

我们考虑从一个给定的\（2 \）-在度量空间中对有限集进行聚类来选择\（p \）点的子集的一个极端问题的计算复杂性。从某些几何准则的角度来看，选定的点子集必须以最佳方式描述给定的簇。这是数据挖掘应用问题的形式化形式，其中包括根据竞争者相似性函数找到由两个类别组成的数据集的典型代表的子集。该问题被证明是NP难的。为此，我们从多项式上将问题从一个很强的意义上简化为NP-hard问题，即\（p \）-中值问题。