It is proved that the k-means++ algorithm is an $O(1)$-approximation with constant probability in this budget-smoothed setup.
Abstract
The $k$-means++ algorithm is a standard and widely used seeding method for $k$-means clustering, but for a fixed number $k$ of centers its worst-case expected approximation ratio is $\Theta(\log k)$. We consider the same algorithm when an adversary first fixes the dataset and some $K$; the number of centers $k$ is then chosen uniformly from $\{K,\ldots,2K-1\}$. We prove that $k$-means++ is an $O(1)$-approximation with constant probability in this budget-smoothed setup.
An expected approximation guarantee of an expected approximation guarantee of $8(\ln k+2)\left(\frac{1+\varepsilon}{1-\varepsilon}{1-\varepsilon}\right)^4 = (1+O(\varepsilon))\,8(\ln k+2)$.
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