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

TL;DR

This paper introduces a primal-dual algorithm for k-means clustering that improves approximation guarantees by leveraging geometric structure and strict cluster constraints, surpassing previous local search methods.

Contribution

It presents a novel primal-dual approach that exploits geometric properties and enforces cluster constraints, achieving a 6.357-approximation for k-means, improving over prior guarantees.

Findings

Achieves a 6.357-approximation for k-means.

Extends techniques to non-Euclidean k-means and Euclidean k-median.

Provides a general framework for clustering approximation guarantees.

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 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) to 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 LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of $k$-means where the underlying metric is not required to be Euclidean and for $k$-median in Euclidean metrics.