This paper investigates why and when the edge-based districting problem becomes computationally intractable. The overall problem is represented as an exact mathematical programming formulation consisting of an objective function and several constraint groups, each enforcing a well-known districting criterion such as balance, contiguity, or compactness. While districting is known to be NP-hard in general, we study what happens when specific constraint groups are relaxed or removed. The results identify precise boundaries between tractable subproblems (in P) and intractable ones (NP-hard). The paper also discusses implications on node-based analogs of the featured districting problems, and it considers alternative notions of certain criteria in its analysis.

翻译：本文研究了基于边的区域划分问题在何种情况下以及为何变得计算上不可处理。该整体问题被表述为一个精确的数学规划模型，包含一个目标函数和多个约束组，每个约束组强制执行一个众所周知的区域划分准则，如平衡性、连通性或紧凑性。尽管区域划分问题通常被认为是NP难的，但我们研究了当特定约束组被放宽或移除时会发生什么。研究结果精确界定了可处理子问题（属于P类）与不可处理子问题（NP难类）之间的边界。本文还讨论了所研究区域划分问题的节点类比问题的影响，并在分析中考虑了某些准则的替代定义。