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