A simple D^2-sampling based PTAS for k-means and other Clustering Problems

Given a set of points $P \subset \mathbb{R}^d$, the $k$-means clustering problem is to find a set of $k$ {\em centers} $C = \{c_1,...,c_k\}, c_i \in \mathbb{R}^d,$ such that the objective function $\sum_{x \in P} d(x,C)^2$, where $d(x,C)$ denotes the distance between $x$ and the closest center in $C$, is minimized. This is one of the most prominent objective functions that have been studied with respect to clustering. $D^2$-sampling \cite{ArthurV07} is a simple non-uniform sampling technique for choosing points from a set of points. It works as follows: given a set of points $P \subseteq \mathbb{R}^d$, the first point is chosen uniformly at random from $P$. Subsequently, a point from $P$ is chosen as the next sample with probability proportional to the square of the distance of this point to the nearest previously sampled points. $D^2$-sampling has been shown to have nice properties with respect to the $k$-means clustering problem. Arthur and Vassilvitskii \cite{ArthurV07} show that $k$ points chosen as centers from $P$ using $D^2$-sampling gives an $O(\log{k})$ approximation in expectation. Ailon et. al. \cite{AJMonteleoni09} and Aggarwal et. al. \cite{AggarwalDK09} extended results of \cite{ArthurV07} to show that $O(k)$ points chosen as centers using $D^2$-sampling give $O(1)$ approximation to the $k$-means objective function with high probability. In this paper, we further demonstrate the power of $D^2$-sampling by giving a simple randomized $(1 + \epsilon)$-approximation algorithm that uses the $D^2$-sampling in its core.