AI 中文总结
研究p区域问题，利用p区域目标与k划分问题的联系提出新ILP模型ER-S，用特定子巡回消除不等式强化已知模型Tree，组合二者得到ER-S-Tree，其在多面体强度上优于现有模型，能解决此前棘手的欧洲国家问题实例。
AI 中文摘要
区域划分是空间分析中的基本任务，旨在将较大区域划分为关于给定属性同质的较小区域。流行的p区域问题模型中，区域由输入平面细分的区域分组形成。给定细分的邻接图G和顶点间成对差异，目标是将G划分为固定数量p的连通子图，以最小化同一子图中所有顶点对的差异之和。该问题是NP难的，即使小实例也难以求解到可证明的最优解。本文提出用于p区域问题的新ILP模型ER-S，利用p区域目标与k划分问题之间的联系。此外，我们用一种特定于p区域问题的新型子巡回消除不等式强化已知的ILP模型Tree。将ER-S和强化版的Tree相结合产生模型ER-S-Tree，其在多面体强度上优于现有模型。这一理论优势在我们的实验评估中体现为其优越的性能。特别是，新模型ER-S和ER-S-Tree能够解决以前难以处理的主要欧洲国家的问题实例。
英文摘要
Regionalization is a fundamental task in spatial analysis that seeks to partition a larger area - such as a country - into smaller regions that are homogeneous with respect to a given attribute. A popular model for regionalization is the p-regions problem, in which regions are formed by grouping the areas of an input planar subdivision. Given the subdivision's adjacency graph G and pairwise dissimilarities between vertices, the goal is to partition G into a fixed number p of connected subgraphs, such as to minimize the sum of dissimilarities over all vertex pairs in the same subgraph. The problem is NP-hard and even small instances are difficult to solve to provable optimality. In this paper, we present the new ILP model ER-S for the p-regions problem, exploiting a connection between the p-regions objective and the k-partitioning problem. Furthermore, we strengthen the known ILP model Tree with a new type of subtour elimination inequality specific to the p-regions problem. Combining ER-S and the strengthened version of Tree yields the model ER-S-Tree, which dominates the state-of-the-art models in polyhedral strength. This theoretical advantage is reflected in its superior performance in our experimental evaluation. In particular, the new models ER-S and ER-S-Tree enable the solution of problem instances for major European countries that were previously intractable.
