A bi-criteria approximation algorithm for $k$ Means

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $βk > k$ clusters, and must produce a clustering with cost at most $α$ times the to the cost of the optimal set of $k$ clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee $α(β)$ depending on the number $βk$ of clusters that may be opened. Our gurantee $α(β)$ is always at most $9 + ε$ and improves rapidly with $β$ (for example: $α(2)<2.59$, and $α(3) < 1.4$). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.