The approximability of different NP-hard problems

Question

I'm fairly new to the topic Computational Complexity and had the following question (I therefore apologies before hand for any poorly stated terminology).

Suppose i have two optimization problems $A$ and $B$ and suppose further that $A$ is polynomial time reducible to $B$ i.e $A \leq_{p} B$. Therefore i know if $A$ is NP-hard that $B$ is also NP-hard.

If however i have more information about $A$ for example

1.$A$ is APX-hard does that also mean that $B$ is APX-hard?

2.$B$ admits a PTAS does that also mean that $A$ admits a PTAS or can it be APX-hard?

Thanks

Answer 1

In general APX-hardness of $A$ will not immediately imply APX-hardness of $B$. Nevertheless, the implication holds if you strengthen the conditions of the reduction. If the reduction is approximation reserving, i.e. an approximation solution to $B$ corresponds to an approximation solution to $A$ when the reduction is applied, the implication still holds. The wikipedia article on approximation-preserving reductions might give you a more formal idea of this concept.