Is Asymptotic PTAS $\subseteq$ APX? The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows:


  A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an algorithm $A_\epsilon$ and $N_\epsilon &gt; 0$, such that $A_\epsilon(I) \leq (1+\epsilon)OPT$ for $OPT(I) \geq N_\epsilon$.


I have two questions:


Is Asymptotic PTAS contained in APX?
If problem $\Pi$ is APX-Hard, does it imply that $\Pi$ is not Asymptotic PTAS (suppose $P \neq NP$)?


I know above are true if we replace Asymptotic PTAS with PTAS (which are common-sense now). But I can't figure out how the definitions of Asymptotic PTAS and APX give answers to above questions..
Both statements seem to be false.


Consider all Boolean formulas $\Phi$ in 3-CNF. The objective value of an $n$-variable formula $\Phi$ is $1/n$ if $\Phi$ is satisfiable, and is $1$ if $\Phi$ is not satisfiable. The corresponding minimization problem has an asymptotic PTAS (it is easy to come within an additive $+1$), but is not in APX (as this would allow you to solve 3-SAT).
Consider all undirected graphs. The objective value of a graph $G$ is its edge-chromatic number $\chi'(G)$. Holyer has proved that deciding $\chi'(G)=3$ is NP-hard. The corresponding minimization problem is APX-hard (there is no approximation algorithm with worst case ratio better than $4/3$), but the problem has an asymptotic PTAS (Vizing's theorem tells us that $\chi'(G)$ is between the max-degree and the max-degree plus 1).