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

This paper studies the k-means++ algorithm for clustering as well as the class of D^ sampling algorithms to which k-means++ belongs. It is shown that for any constant factor $β> 1$, selecting $βk$ cluster centers by D^ 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 $β= 1$ to the constant-factor bi-criteria regime. It also improves upon an existing constant-factor bi-criteria result that holds only with constant probability.