A Constant-Factor Bi-Criteria Approximation Guarantee for k-means plus

This paper studies the k-means++ algorithm for clustering as well as the class of D-l sampling algorithms to which k-means++ belongs. It is shown that for any constant factor beta > 1, selecting beta k cluster centers by D-l sampling yields a constant-factor approximation to the optimal clustering with k centers, in expectation and without conditions on the dataset. This result extends the previously known O(log k) guarantee for the case beta = 1 to the constant-factor bi-criteria regime. It also improves upon an existing constant-factor bi-criteria result that holds only with constant probability.