Let's say I'm given two set cover problems 1 and 2. In both problems I'm required to cover the same universe $U$. For each of the set cover problems, I have sets $S_{1,i}.....S_{1,n}$ and $S_{2,j}.....S_{2,m}$ with corresponding weights. Assume both are well-posed. I want to know how their optimal values compare, and their dependence on the input data.
Has there been research done regarding the dependence of the set cover problem on the sets $S$? (data dependence). What keywords to search that might be relevant for this?