Local Search Yields a PTAS for $k$-Means in Doubling Metrics

The most well-known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centers and partition the data points into $k$ clusters around these centers so that the sum of squares of distances of the points to their cluster center is minimized. Typically these data points lie in Euclidean space $\mathbb{R}^d$ for some $d\geq 2$. $k$-means and the first algorithms for it were introduced in the 1950s. Over the last six decades, hundreds of papers have studied this problem and different algorithms have been proposed for it. The most commonly used algorithm in practice is known as Lloyd--Forgy, which is also referred to as “the” $k$-means algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [Comput. Geom., 28 (2004), pp. 89--112] analyzed a very simple local search heuristic to get a polynomial-time algorithm with approximation ratio $9+\epsilon$ for any fixed $\epsilon>0$ for $k$-means in Euclidean space. Finding an algorithm with a better worst-case approximation guarantee has remained one of the biggest open questions in this area, in particular, whether one can get a true polynomial-time approximation scheme (PTAS) for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for $k$-means for $\mathbb{R}^d$ for any fixed $d$. More precisely, for any error parameter $\epsilon>0$, the local search algorithm that considers swaps of up to $\rho=d^{O(d)}\cdot{\epsilon}^{-O(d/\epsilon)}$ centers at a time will produce a solution using exactly $k$ centers whose cost is at most a $(1+\epsilon)$-factor greater than the optimum solution. Although the algorithm is not practical due to the large polynomial running time, it settles the approximability of this important problem. Our analysis extends very easily to the more general settings where we want to minimize the sum of $q$th powers of the distances between data points and their cluster centers (instead of sum of squares of distances as in $k$-means) for any fixed $q\geq 1$ and where the metric may not be Euclidean but still has fixed doubling dimension. Finally, our techniques also extend to other classic clustering problems. We provide the first demonstration that local search yields a PTAS for uncapacitated facility location and the generalization of $k$-median to the setting with nonuniform opening costs in doubling metrics.