Minimum capacity cut reduction from digraph with two edge weight sets
This problem is NP-hard. 

Consider the $f=0$ case, then you are trying to minimize $-\Sigma_{e\in\delta^-(X)}g(e)$, i.e. maximize $\Sigma_{e\in\delta^-(X)}g(e)$. This is exactly the maximum cut problem, which is NP-hard.