  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.
