k-means++: few more steps yield constant approximation
This provides a more efficient method for achieving constant approximation in k-means clustering, which is incremental but addresses a specific bottleneck in clustering algorithms.
The paper tackles the problem of improving the approximation guarantee for k-means clustering by showing that adding only εk local search steps to the k-means++ algorithm yields a constant approximation with high probability, resolving an open problem from prior work.
The k-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is a state-of-the-art algorithm for solving the k-means clustering problem and is known to give an O(log k)-approximation in expectation. Recently, Lattanzi and Sohler (ICML 2019) proposed augmenting k-means++ with O(k log log k) local search steps to yield a constant approximation (in expectation) to the k-means clustering problem. In this paper, we improve their analysis to show that, for any arbitrarily small constant $\eps > 0$, with only $\eps k$ additional local search steps, one can achieve a constant approximation guarantee (with high probability in k), resolving an open problem in their paper.