Inapproximability of an optimization problem

Question

Suppose we have an optimization problem $\mathcal{P}$ that we should cover all points with $k$ disjoint rectangles in the plane and we should optimize a distance function over each rectangles . Now, suppose there is a $\mathcal{P}'$ that just need cover all points in the plane with $k$ disjoint rectangles.

Already proved that $\mathcal{P}'$ is NP-Hard and there is no constant factor approximation algorithm for $\mathcal{P}'$. Can we conclude that $\mathcal{P}$ has no constant factor approximation algorithm? Why?

I think as follow:

$\mathcal{P}$ is at least hard as $\mathcal{P}'$ so if there is a constant factor approximation algorithm for $\mathcal{P}$ then for each feasible solution $\mathcal{I}$ of $\mathcal{P}$, then $\mathcal{I}$ is a solution for decision version of $\mathcal{P}'$ hence we solve decision version of $\mathcal{P}'$ in polynomial time and hence $P=NP$. Finally, we conclude that $\mathcal{P}$ has no constant factor approximation algorithm.

Answer 1

Here is a similar example. Let $\mathcal{Q}$ be the problem of maximizing the number of satisfied clauses in an input CNF given that the assignment is the FALSE assignment. Let $\mathcal{Q'}$ be the problem of maximizing the number of satisfied clauses in an input CNF, without any other constraints. It is known that $\mathcal{Q'}$ has no PTAS (unless P=NP). Does it follow that $\mathcal{Q}$ has no PTAS?