Finding Regions of Maximum Circularity in Plane Geometric Graphs

2026-07-30

AI Summary

This study addresses the problem of selecting a subset of faces from a given planar polygonal partition to maximize α-roundness, defined as the ratio of area to the α-th power of perimeter for α > 1—a measure relevant to generating compact regions and combating gerrymandering in geographic information science. The authors establish, for the first time, that this optimization problem is weakly NP-hard when α ∈ (1, 2]. Furthermore, they present a pseudo-polynomial time algorithm applicable to any α > 1. By integrating techniques from computational geometry, combinatorial optimization, and dynamic programming, this work delineates the computational complexity landscape of the α-roundness maximization problem and provides an efficient algorithmic framework for its solution.

Abstract

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 $\frac{4πA}{P^2}$, where $A$ is the area of the selected shape and $P$ is its perimeter. We consider the more general task of maximizing $\frac{A}{P^α}$ for an arbitrary $α>1$, which we call the $α$-circularity problem. We perform the first rigorous study of its complexity and show that it is weakly NP-hard if $α\in (1,2]$. Furthermore, for $α>1$ we present a pseudopolynomial time algorithm for this problem.

Problem

• circularity
• Polsby-Popper score
• compact regions
• geometric graph
• polygonal subdivision

Innovation

• circularity optimization
• Polsby-Popper score
• weakly NP-hard
• pseudopolynomial algorithm
• geometric graph