Probably a more theoretical way is to prove that your problem has a Matroid structure. If you can prove that your problem has such a structure, there is a greedy algorithm to solve it.
According to the classic book Introduction to Algorithms, matroid a is an ordered pair M = (S, l) with:
* S is a finite nonemtpy set * l is a nonempty family of subsets of S, such that B element of l and A a subset of B than also A is element of l. l is called the independent subsets of S. * If A and B are elements of l and A is a smaller cardinality than B, then there is some element x that is in B, but not in A, so that A extended by x is also element of l. That is called exchange property.
Often there is also a weight function w, which assigns a weight to each element x in S.
If you can formulate your function as a weighted matroid, then the following pseudo-code, like Python, solves your problem:
GREEDY(M,w): (S,l) = M a = {} sort S into monotonically decreasing order by weight w for x in S: if A + {x} in l: A = A + {x}