AI 中文总结
本文研究平面几何图中最大化α-圆度的问题，证明α∈(1,2]时该问题弱NP难，针对任意α>1提出伪多项式时间算法，为地理信息科学的区域紧凑性生成提供理论支撑。
AI 中文摘要
地理信息科学的不同应用场景中存在一个问题：从地图上的区域生成紧凑区域，这在选区划分中避免不公正操纵选区划分（gerrymandering）方面尤为重要。区域紧凑性的常用衡量标准是波尔兹比-波珀（Polsby-Popper）分数，该分数基于区域面积A和周长P衡量给定区域与圆的接近程度，公式为$\frac{4πA}{P^2}$。本文假设给定平面的多边形细分，研究选择多边形面子集以最大化波尔兹比-波珀分数的问题；进一步考虑更通用的任务，即最大化$\frac{A}{P^α}$（其中α>1），将此问题称为α-圆度问题。本文首次对该问题的复杂性进行严格研究，证明当α∈(1,2]时，该问题是弱NP难的；此外，对于任意α>1，本文提出了该问题的伪多项式时间算法。
英文摘要
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.
