k-means++: few more steps yield constant approximation

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. 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 epsilon > 0, with only epsilon * k additional local search steps, one can achieve a constant approximation guarantee (with high probability in k), resolving an open problem in their paper.