  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.
