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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Consider $k$ independent Bernoulli populations, each sampled $n$ times, and select the $t$ populations with the largest success counts, breaking ties uniformly. Classical monotonicity reduces the worst case over the preference zone with separation $\delta$ to the slippage family with levels $p$ and $p+\delta$, leaving...
After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.
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