I am considering the following problem: there is a set of elements $S$ where each element is assigned to a bin $B$. The bins are disjoint and their union is $S$. There is also a cost function assigning each element $e \in S$ a cost $c$. The problem is to maximize the number of different bins and minimize the total cost of the elements, subject to a cardinality constraint $K$. (I'm interested in both cases: either at most or exactly $K$ elements). So the objective function would be: $f(S) = \lambda\sum_{i=1}^N \mathbb{1}_{|B_{i} \cap S| > 0} - \sum_{e \in S} c(e)$, where $N$ is the number of bins and $\lambda$ is a tradeoff parameter. Maximizing the number of different bins is equivalent to maximizing the rank of a partition matroid. In addition to partition matroid rank, I'm also interested in other types of coverage-like submodular functions. This can also be seen as maximizing the difference between a sub modular function and a modular cost function; it is non monotone sub modular. Is this - or a similar formulation - equivalent to any other known problem? 
Maximize number of bins and minimize cost of elements chosen from a set
ds.algorithms;co.combinatorics;approximation algorithms;packing;submodularity