Before revision:  The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for the same class of geometric objects .
After revision:  The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system (X, \mathcal{S .