A Complexity Theoretical Study of Fuzzy K-Means

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.