Approximation algorithms for NP-complete problems

Given two NP NP-hard functional problems, A and B, one can find a reduction of A to B. Is it possible to find a reduction that would honour approximations? That is, if you have an approximation algorithm for B that yield approximate solutions within accuracy $\delta$, is it possible to reduce A to B in such a way that one would be able to derive an approximate solution of A within accuracy $\epsilon = \epsilon(\delta)$?

• I'll leave a proper answer to experts, but let me point you towards the notion of strong np-completeness. – Raphael Dec 16 '13 at 20:07
• In certain circumstances an approximation-preserving reduction is possible. These instances are characterized by L-reductions. However, as @Raphael, noted, not all problems possess this feature. – Nicholas Mancuso Dec 17 '13 at 17:05