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Strong ILP Formulations for the p-Regions Problem

Authors Daniel Faber , Jan-Henrik Haunert , Petra Mutzel

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Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
Keywords
  • p-regions problem
  • connected graph partitioning
  • area aggregation
  • integer linear programming
  • branch-and-cut

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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.

Cite As Get BibTex

Daniel Faber, Jan-Henrik Haunert, and Petra Mutzel. Strong ILP Formulations for the p-Regions Problem. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.13

Author Details

Daniel Faber
  • University of Bonn, Germany
Jan-Henrik Haunert
  • University of Bonn, Germany
Petra Mutzel
  • University of Bonn, Germany
  • Lamarr Institute, Bonn, Germany

Funding

This research was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant FOR-5361 - 459420781.

Acknowledgements

We want to thank Michael Kaibel for his helpful remarks.

Supplementary Materials

References

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