In the Set Cover problem we need to cover each item at least once.
I am considering the case in which I want each item to be covered at least $ k $ times with constant $ k $.
I consider the classic LP for the problem and random rounding.
Is it true that the modification of the LP of $ geq 1 $ to $ geq k $ in the coverage restriction and with the same rounding (up to the number of repetitions) also works well for this variant?
It seems so, but I'm not sure if I'm missing something.