Let's say I have an optimization problem called $k$-foo which asks for a solution of size $k$ minimizing some quality criterion.
Now the corresponding decision problem $foo(M)$ would be:
Is there a solution to foo with quality at least $M$ of size $k$.
For problems on one parameter (for example vertex cover) it is obvious that solving the optimization problem sovles the decision problem.
But here I do not see such a correspondance between the $k$-foo optimization problem and the $foo(M)$ decision problem. How does for example showing that $foo(M)$ is NP-hard implies that $k$-foo is NP-hard?
The $k$-center problem is an example of such a problem where the decision version takes the radius as input and asks wether a solution of size $k$ exists.