Strong ILP Formulations for the p-Regions Problem

2026-07-06

AI Summary

This study addresses the p-regions problem in geographic zoning, which involves partitioning a region into p connected and attribute-homogeneous subregions—a task known to be NP-hard. To tackle this challenge, we propose a novel integer linear programming model, ER-S, enhanced with specialized subtour elimination inequalities that strengthen connectivity constraints. Building upon structural insights from the k-partitioning problem, we further develop the ER-S-Tree model, which exhibits superior polyhedral strength. Computational experiments demonstrate that our approach significantly outperforms existing methods in both computational efficiency and provable optimality, successfully solving large-scale instances for several European countries for the first time.

Abstract

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.

Problem

• p-regions problem
• regionalization
• connected partitioning
• NP-hard optimization
• spatial analysis

Innovation

• p-regions problem
• integer linear programming
• subtour elimination
• regionalization
• k-partitioning