# largest bitwise-or subset with a maximum number of on bits?

Suppose I have the following set of bit sets:

 [1,]    0    1    1    1    0    1    0    1    0     0
[2,]    1    0    1    1    0    0    0    1    0     1
[3,]    0    0    1    0    1    1    0    1    1     0
[4,]    1    0    1    0    0    0    0    1    1     1
[5,]    0    1    0    1    0    1    0    1    1     0
[6,]    0    1    1    1    1    0    0    1    0     0
[7,]    1    1    0    0    0    1    1    0    0     1
[8,]    1    1    0    1    1    1    0    0    0     0
[9,]    0    1    1    0    1    1    0    0    0     1
[10,]    0    1    1    1    0    0    1    1    0     0


How would I go about finding the largest set such that the bitwise-or of that set does not exceed n bits in the "on" position.

For example, if I or rows 1 & 10...

 [1,]         0    1    1    1    0    1    0    1    0     0
[10,]         0    1    1    1    0    0    1    1    0     0
[1 or 10,]    0    1    1    1    0    1    1    1    0     0


This gives me 7 bits in the "on" position.

Now suppose my limit was 8 bits... how would I go about finding the largest set?

Edit: I know how to do this as a mixed-integer-linear program, but I'm assuming there's a nifty graph approach.

This problem is NP-hard. It is called "the minimum-k-union problem". Given a universe $$U$$ and a family of sets $$\mathcal{F}$$ over this universe. Are there $$k$$ distinct sets in $$F$$ such that the union of all these sets has size at most $$d$$.
Your presentation is the matrix formulation of this problem where each column corresponds to an element in the universe and each row corresponds to a set in the family. We are looking for $$k$$ rows such that the number of columns having at least one "1" is at most $$d$$.