Improved Approximations for Euclidean $k$-means and $k$-median, via Nested Quasi-Independent Sets

TL;DR

This paper presents a new primal-dual algorithm for Euclidean $k$-median and $k$-means problems, achieving improved approximation ratios by leveraging a novel nested quasi-independent set technique.

Contribution

The authors introduce a primal-dual algorithm with better approximation ratios and a new nested quasi-independent set method for Euclidean clustering.

Findings

Achieved approximation ratio of 2.406 for Euclidean $k$-median.

Achieved approximation ratio of 5.912 for Euclidean $k$-means.

Introduced a new technique of nested quasi-independent sets for optimization.

Abstract

Motivated by data analysis and machine learning applications, we consider the popular high-dimensional Euclidean $k$-median and $k$-means problems. We propose a new primal-dual algorithm, inspired by the classic algorithm of Jain and Vazirani and the recent algorithm of Ahmadian, Norouzi-Fard, Svensson, and Ward. Our algorithm achieves an approximation ratio of $2.406$ and $5.912$ for Euclidean $k$-median and $k$-means, respectively, improving upon the 2.633 approximation ratio of Ahmadian et al. and the 6.1291 approximation ratio of Grandoni, Ostrovsky, Rabani, Schulman, and Venkat. Our techniques involve a much stronger exploitation of the Euclidean metric than previous work on Euclidean clustering. In addition, we introduce a new method of removing excess centers using a variant of independent sets over graphs that we dub a "nested quasi-independent set". In turn, this technique may be of interest for other optimization problems in Euclidean and $\ell_p$ metric spaces.