Better Guarantees for $k$-Means and Euclidean $k$-Median by Primal-Dual Algorithms

Abstract

Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for $k$-median in Euclidean metrics and for a generalization of $k$-means in which the underlying metric is not required to be Euclidean.