I want to solve the following problem :
For two set $\mathcal N$ and $\mathcal E$, we have a binary matrix $A$ index by $\mathcal N \times \mathcal E$ and we say that $e\in \mathcal E$ covers $n \in \mathcal N$ if $a_{n, e}= 1$. We are looking for the maximum set of $\mathcal N$ covered by exactly $k$ elements in $\mathcal E$.
I know that we can write this problem as an integer program. However, I want to have a combinatorial algorithm to solve it. Can anyone help me to find good heuristic or an exact algorithm?
As I understood, you are asking about so-called Maximum Coverage problem. It is so well known and investigated, that I can easily provide you a very brief survey on it, containing a list of its applications, results of its computational complexity and approximability, related problems and more easy special cases. Namely, this is the beginning of the introduction of the paper “Approximation Schemes for Geometric Coverage Problems” by Steven Chaplick, Minati De, Joachim Spoerhase, and me. :-)