Approximation preserving reductions for set covering, vertex covering and independent set hierarchies under differential approximation

The notion of approximability preserving reductions between different problems deserves special attention in approximability theory. These kinds of reductions allow us polynomial time conversion of some already known 'good' approximation algorithms for some NP-hard problems into ones for some other NP-hard problems. In this context, we consider reductions for set covering and vertex covering hierarchies. Our results are then extended to hitting set and independent set hierarchies. Here, we adopt the differential approximation ratio that has the natural property to be stable under affine transformations of the objective function of a problem.ou