A lot of approximation algorithms are based on relaxation. The way it usually works is this. You take the original problem and relax it some large class of efficiently solvable problem (e.g. relax an IP to an LP). Any solution to the original problem of cost $X$ gives a corresponding solution to the relaxation of cost $X$.
You then convert a solution to the relaxed problem of cost $Y$ to a solution to the original problem of cost $\alpha Y$ for some $\alpha > 1$. This implies that you have an $\alpha$ approximation.
But what if you could argue that the original feasible solution of cost $X$ guarantees the existence of a solution to the relaxation of cost $X/\beta$? Then you would get an $\alpha/\beta$ approximation algorithm.
Has this strategy ever been employed? Is there ever a case when you can show that a feasible solution to the original problem corresponds to a feasible solution to the relaxation of strictly lower cost ?