In the Set Cover problem we need to cover each element at least once. I'm considering the case where I want each element to be covered at least $k$ times with constant $k$.
I consider the classic LP for the problem and randomized rounding. Is it indeed the case that the modification of the LP from $\geq 1$ to $\geq k$ in the covering constraint and with the same rounding (up to the number of repetitions) works well for this variant as well?
It looks like it, but I'm not sure if I'm missing anything.