Suppose a set function $f(S)$ is submodular and non-increasing, meaning that for any $S'\subset S$, $f(S') \geq f(S)$. The problem is to minimize $f(S)$ s.t. $|S| \leq k$.
I am wondering if there are some references discussing approximation algorithms regarding this problem.
The reference I can find now discussed a similar problem: minimize a non-decreasing submodular function under (lower bound) cardinality constraint (i.e., $|S| \leq k$). But it is not obvious to me how to adapt the result to the previous problem.
Edit: the exact form of $f$ is known.