  The k-means++ algorithm due to Arthur and Vassilvitskii has become the most
popular seeding method for Lloyd's algorithm. It samples the first center
uniformly at random from the data set and the other $k-1$ centers iteratively
according to $D^2$-sampling where the probability that a data point becomes the
next center is proportional to its squared distance to the closest center
chosen so far. k-means++ is known to achieve an approximation factor of $O(\log
k)$ in expectation. Already in the original paper on k-means++, Arthur and
Vassilvitskii suggested a variation called greedy k-means++ algorithm in which
in each iteration multiple possible centers are sampled according to
$D^2$-sampling and only the one that decreases the objective the most is chosen
as a center for that iteration. It is stated as an open question whether this
also leads to an $O(\log k)$-approximation (or even better). We show that this
is not the case by presenting a family of instances on which greedy k-means++
yields only an $\Omega(\ell\cdot \log k)$-approximation in expectation where
$\ell$ is the number of possible centers that are sampled in each iteration. We
also study a variation, which we call noisy k-means++ algorithm. In this
variation only one center is sampled in every iteration but not exactly by
$D^2$-sampling anymore. Instead in each iteration an adversary is allowed to
change the probabilities arising from $D^2$-sampling individually for each
point by a factor between $1-\epsilon_1$ and $1+\epsilon_2$ for parameters
$\epsilon_1 \in [0,1)$ and $\epsilon_2 \ge 0$. We prove that noisy k-means++
compute an $O(\log^2 k)$-approximation in expectation. We also discuss some
applications of this result.
