The fuzzy K -means problem is a popular generalization of the well-known K -means problem to soft clusterings. In this article, we present the first algorithmic study of the problem going beyond heuristics. Our main result is that, assuming a constant number of clusters, there is a polynomial time approximation scheme for the fuzzy K -means problem. As a part of our analysis, we also prove the existence of small coresets for fuzzy K -means. At the heart of our proofs are two novel techniques developed to analyze the otherwise notoriously difficult fuzzy K -means objective function.

中文翻译：

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模糊 K 均值的复杂性理论研究

模糊的ķ-均值问题是众所周知的普遍问题ķ- 表示软聚类的问题。在本文中，我们提出了超越启发式的问题的第一个算法研究。我们的主要结果是，假设一个恒定数量的簇，有一个多项式时间近似方案用于模糊ķ- 表示问题。作为我们分析的一部分，我们还证明了用于模糊的小核心集的存在ķ-方法。我们证明的核心是开发了两种新技术，用于分析其他众所周知的困难模糊ķ- 表示目标函数。