Given positive integers $k$, $m$, $n$, with $m,n >> k$, suppose we have $n$ boxes each containing $k$ randomly (uniformly) selected positive integers $x$ satisfying $1 \leq x \leq m$ (duplicates in the box are permitted).

I begin selecting distinct positive integers $y$ such that $1 \leq y \leq m$ until one integer from each box has been selected. Call a box "marked" if at least one integer in the box has been chosen. I am interested in the expected number of $y$'s that need to be chosen until each box has been marked. At each step, a $y$ contained in the largest number of unmarked boxes is chosen.

This seems similar to some problems related to computer hashing functions (which is where it arose, though at this point the connection is a little tenuous), but I have been unable to find precisely this problem. I wonder if anyone knows the problem or sees a quick solution?