Approximation quality of minimum k-cut

Question

Can someone please clarify a point of confusion regarding approximate solutions to the minimum k-cut problem.

In the wiki page of Minimum K-cut, it is stated that:

Several approximation algorithms exist with an approximation of $2-2/k.$

Other than the Gomory Hu tree based algorithm, a simple greedy algorithm can be found in page 7 here which also has the approximation ratio $2-2/k$.

My question is: is this approximation factor respect to the optimal solution when $k$ is considered as fixed or when it is part of the input?

Answer 1

In the slides that you refer to, it is stated that:

"The minimum k-cut problem is polynomial time solvable for fixed k; however, it is NP-hard if k is specified as part of the input"

So yes, the approximation is only needed for the case when $k$ is part of the input.