Consider the following min-max problem
Given a graph $G=(V,E)$ and an integer $k \geq 0$, delete at most $k$ nodes in $G$ to maximize the size of the minimum dominating set in the residual graph.
The NP-hardness of the problem can be shown by setting $k=0$. Can we somehow easily get the hardness of approximation results from the hardness of the Dominating Set problem? In general, for min-max problems, can we do the same thing?