Can we multiply approximation hardness?
When we say that a minimization problem A is NP-hard to approximate up to a factor $\alpha \geq 1$, we mean that there is a polynomial time reduction $f$ from instances $\varphi$ of SAT to instances $x$ of A annotated by a target value $\theta$ such that:


If $\varphi$ is satisfiable then $A(x) \leq \theta$ (where $A(x)$ is the optimal value of $x$).
If $\varphi$ is not satisfiable then $A(x) &gt; \alpha \theta$.


Using this definition it is not hard to prove the multiplicative property.



You can also consider a more general notion of hardness: if there is a polynomial time $\alpha$-approximation algorithm for A, then P=NP. Even under this notion, a $\alpha$-hardness for A translates to a $\alpha^2$-hardness for "A2". Indeed, given an instance of A, duplicate it and pass it to the $\alpha^2$-approximation algorithm. One of the solutions is an $\alpha$-approximation. If only the objective value is given, then the smaller one is an $\alpha$-approximation.