Is there a known approximation algorithm for the problem of minimizing a monotone (non increasing) supermodular function under partition matroid constraints ?
Consider a graph $G=(V,E)$. Define the set function $f$ over $V$ where $f(A)$ is the number of edges with both end points in $A$. Then $f$ is supermodular and monotone. Suppose $G$ has an independent set of size $k$. Then the minimum of $f(S)$ under the constraints $|S| = k$ (the uniform matroid constraint) will be $0$. If $G$ does not have an independent set of size $k$ then the min will be at least $1$. Hence the problem is inapproximable.