Complexity and Approximation of the Fuzzy K-Means Problem

The fuzzy K-means problem is a generalization of the classical K-means problem to soft clusterings, i.e. clusterings where each points belongs to each cluster to some degree. Although popular in practice, prior to this work the fuzzy K-means problem has not been studied from a complexity theoretic or algorithmic perspective. We show that optimal solutions for fuzzy K-means cannot, in general, be expressed by radicals over the input points. Surprisingly, this already holds for very simple inputs in one-dimensional space. Hence, one cannot expect to compute optimal solutions exactly. We give the first (1+ ǫ)-approximation algorithms for the fuzzy K-means problem. First, we present a deterministic approximation algorithm whose runtime is polynomial in N and linear in the dimension D of the input set, given that K is constant, i.e. a polynomial time approximation algorithm given a fixed K. We achieve this result by showing that for each soft clustering there exists a hard clustering with comparable properties. Second, by using techniques known from coreset constructions for the K-means problem, we develop a deterministic approximation algorithm that runs in time almost linear in N but exponential in the dimension D. We complement these results with a randomized algorithm which imposes some natural restrictions on the input set and whose runtime is comparable to some of the most efficient approximation algorithms for Kmeans, i.e. linear in the number of points and the dimension, but exponential in the number of clusters.