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

TL;DR

This paper improves the analysis of the k-means++ algorithm by showing that a small number of additional local search steps can achieve a constant approximation ratio with high probability, advancing clustering efficiency.

Contribution

The authors demonstrate that adding only εk local search steps to k-means++ suffices for a constant approximation, resolving an open problem in clustering algorithms.

Findings

Achieves constant approximation with high probability

Requires only εk additional local search steps

Improves upon previous O(log k) approximation guarantees

Abstract

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.