# Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is a closed chain of straight line segments that does not cross itself).

I am interested in the complexity of this problem:

Input: Set $$S$$ of $$N$$ points on integer 2D grid and a rational number $$A$$

Question: Is there a polygonalization that has enclosed area $$A$$?

Is this problem still NP-complete?

The area $$A$$ is given as an integer (represented in unary)

Fekete, Sándor P. "On simple polygonalizations with optimal area." Discrete & Computational Geometry 23.1 (2000): 73-110

Instance: a set $$S$$ of $$n$$ grid points in the plane
Question: Is there a simple polygon on this vertex set $$S$$ that does not contain any other grid points on its boundary or in its interior?
By Pick's theorem, every simple polygon with grid vertices that has $$b$$ grid points on its boundary and $$i$$ grid points in its interior has area $$(b+2i-2)/2$$.
For the scenario in GRID-EMPTY, the smallest possible area is $$(n-2)/2$$, and this area can be reached if and only if the answer to the GRID-EMPTY instance is YES.
Hence, you simple have to set $$A:=(n-2)/2$$ in Fekete's reduction and get NP-completeness of your question.