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arXiv · 2204.04828

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

Vincent Cohen-Addad·Hossein Esfandiari·Vahab Mirrokni·Shyam Narayanan
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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.

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Vincent Cohen-Addad, Hossein Esfandiari, Vahab Mirrokni, Shyam Narayanan. 2022-04-11. Improved Approximations for Euclidean $k$-means and $k$-median, via Nested Quasi-Independent Sets. https://arxiv.org/abs/2204.04828

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