Now I have a set function $f$ (it's monotone), which inputs a node set $S$, and output a real value $r=f(S)$. And I want to compute $$ \max_{S,|S|\leq k}f(S). $$ where k is given number. And I know this problem is NP-hard now.
If f(S) is a submodularity function, then we can use greedy algorithm to compute it with a good approximation in polynomial time. But unfortunately, f(S) is not a submodularity function. So I want to ask is there any other property like sub-modularity to help compute this problem?