Question in coreset construction fro K-median clustering
Of course not. Otherwise the construction would be &quot;pick an arbitrary set of centers&quot;.
You can follows the proof and see that the size of the coreset depends polynomially on $\beta$ and $\alpha$.
If they are $O(1)$ they can be hidden in the $O()$ notation. Otherwise, someone needs to  pay.  See more details in Section 6 here:
https://arxiv.org/abs/2011.09384