Algorithm for Euclidean K-Center Problem

Question

I am studying the Euclidean $k$-center problem. This paper proved that the problem is NP-hard for any arbitrary $k$. However, in this paper authors provided an algorithm for finding solutions for the $k$-center problem in $O(n^{O(\sqrt{k})})$ time. I am confused. If the problem is NP-hard, how we are getting that solution? I don't know whether I am missing something silly here.

Answer 1

If a problem is $NP$-hard, this means that it is (assuming $P$ is not equal to $NP$) impossible to compute the optimal solution in time polynomial in the inputsize.

because $k$ is part of the input the term $n^{O(\sqrt{k})}$ is not considered polynomial, because the exponent is not a constant but depends on the input.