A Refined Approximation for Euclidean k-Means

TL;DR

This paper improves the approximation ratio for the Euclidean k-Means problem from 6.357 to 6.12903 by refining existing analysis, and also establishes a lower bound on the LP's integrality gap.

Contribution

It provides a minor modification to existing analysis to achieve a better approximation ratio and demonstrates a new lower bound on the LP's integrality gap.

Findings

Improved approximation ratio from 6.357 to 6.12903.

Established a lower bound on the LP's integrality gap (>1.2157).

Refined analysis techniques for Euclidean k-Means approximation.

Abstract

In the Euclidean $k$-Means problem we are given a collection of $n$ points $D$ in an Euclidean space and a positive integer $k$. Our goal is to identify a collection of $k$ points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in $D$ and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual $6.357$ approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved $6.12903$ approximation. As a related result, we also show that the mentioned LP has integrality gap at least $\frac{16+\sqrt{5}}{15}>1.2157$.