About a pre-processing step for primal-dual weighted set cover problem
The goal is to give an $O(\log n)$ approximation algorithm, starting with an algorithm that is already an $O(\log n)$ approximation algorithm, and optimizing it further. Adding the low cost elements results in an approximation ratio of $1 + O(\log n) = O(\log n)$, since the total weight of sets we add is $\beta$, which is also an upper bound on the value of the optimal solution.