@inproceedings{2367,
  abstract     = {{One of the most popular fuzzy clustering techniques is the fuzzy K-means algorithm (also known as fuzzy-c-means or FCM algorithm). In contrast to the K-means and K-median problem, the underlying fuzzy K-means problem has not been studied from a theoretical point of view. In particular, there are no algorithms with approximation guarantees similar to the famous K-means++ algorithm known for the fuzzy K-means problem. This work initiates the study of the fuzzy K-means problem from an algorithmic and complexity theoretic perspective. We show that optimal solutions for the fuzzy K-means problem cannot, in general, be expressed by radicals over the input points. Surprisingly, this already holds for simple inputs in one-dimensional space. Hence, one cannot expect to compute optimal solutions exactly. We give the first (1+eps)-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 scheme (PTAS) for fixed K. We achieve this result by showing that for each soft clustering there exists a hard clustering with similar 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 sought solution and whose runtime is comparable to some of the most efficient approximation algorithms for K-means, i.e. linear in the number of points and the dimension, but exponential in the number of clusters.}},
  author       = {{Blömer, Johannes and Brauer, Sascha and Bujna, Kathrin}},
  booktitle    = {{2016 IEEE 16th International Conference on Data Mining (ICDM)}},
  isbn         = {{9781509054732}},
  keywords     = {{unsolvability by radicals, clustering, fuzzy k-means, probabilistic method, approximation algorithms, randomized algorithms}},
  location     = {{Barcelona, Spain}},
  pages        = {{805--810}},
  publisher    = {{IEEE}},
  title        = {{{A Theoretical Analysis of the Fuzzy K-Means Problem}}},
  doi          = {{10.1109/icdm.2016.0094}},
  year         = {{2016}},
}