Finding Regions of Maximum Circularity in Plane Geometric Graphs
A problem that occurs in different applications in geographical information science is to generate compact regions from areas on a map. This is important, e.g., in the context of electoral districting to avoid gerrymandering. A common measure for the compactness of a region is the Polsby-Popper score, which measures how close a given region is to a circle based on its area and perimeter. We assume that a polygonal subdivision of the plane is given and study the problem of selecting a subset of the polygonal faces that maximizes the Polsby-Popper score, given by $\frac{4\pi A}{P^2}$, where $A$ is the area of the selected shape and $P$ is its perimeter. We consider the more general task of maximizing $\frac{A}{P^\alpha}$ for an arbitrary $\alpha>1$, which we call the $\alpha$-circularity problem. We perform the first rigorous study of its complexity and show that it is weakly NP-hard if $\alpha \in (1,2]$. Furthermore, for $\alpha>1$ we present a pseudopolynomial time algorithm for this problem.