[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83862-en":3,"doc-seo-83862-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83862,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Strong ILP Formulations for the p-Regions Problem","Regionalization partitions a larger planar area into smaller homogeneous regions. The p-regions problem forms p connected regions by grouping vertices of an input planar subdivision, using an adjacency graph and pairwise dissimilarities to minimize the sum of dissimilarities across vertex pairs in the same region. The problem is NP-hard and exact optimization is difficult. The paper introduces the ER-S ILP model and strengthens the Tree model with p-specific subtour elimination, combining them as ER-S-Tree.","arXiv :2607 .04886v 1 [ cs .DM] 6 Jul 2026  \nStrong ILP Formulations for the p-Regions Problem  \nDaniel Faber \\#  University of Bonn, Germany Jan-Henrik Haunert \\#  University of Bonn, Germany Petra Mutzel \\#  University of Bonn, Germany  \n~~ Abstract ~~  \nRegionalization 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.  \nIn 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.  \n2012 ACM Subject Classification Theory of computation → Discrete optimization  \nKeywords and phrases p-regions problem, connected graph partitioning, area aggregation, integer linear programming, branch-and-cut  \nDigital Object Identifier 10.4230/LIPIcs.CVIT.2016.23  \nSupplementary Material Code available under [https://github.com/s6dafabe/pRegionsERS](https://github.com/s6dafabe/pRegionsERS)  \nFunding This research was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant FOR-5361 – 459420781.  \nAcknowledgements We want to thank Michael Kaibel for his helpful remarks.  \n 1  Introduction  \nThe aggregation of spatial data is a core topic in geoinformation science, and has given rise to many computationally challenging problems. A common restriction in these aggregation problems is regional connectivity: Given a set of geographic areas, the goal is to group similar (with respect to a given attribute) areas into a smaller number of regions, where each region needs to be contiguous. There are numerous problems that fall into this class of connected clustering problems, such as area aggregation in map generalization [21] and political districting [22] . Almost all of these problems are NP-hard, and solving them to optimality has proven to be a difficult task. Integer Linear Programming (ILP) techniques have shown great success for many difficult combinatorial optimization problems, and thus  \n© Daniel Faber, Jan-Henrik Haunert, and Petra Mutzel;  \nlicensed under Creative Commons License CC-BY 4.0 42nd Conference on Very Important Topics (CVIT 2016) .  \nEditors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:32  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n23:2 Strong ILP Formulations for the p-Regions Problem  \nthe majority of exact algorithms in the area of connected clustering problems are based on ILP formulations.  \nOne instance of an NP-hard connected clustering problem is the p-regions problem, introduced by Duque et al. [9] . They consider the problem of clustering a set of given areas into a predefined number of p spatially contiguous regions, while minimizing the total heterogeneity of all regions. The authors define the heterogeneity of a region by the sum of","cbCaiammbtE3nQDv","https://ap.wps.com/l/cbCaiammbtE3nQDv","pdf",1357955,5,1,32,"English","en",105,"# Introduction\n# Related Work and Connections to k-Partitioning","[{\"question\":\"What is the p-regions problem in the paper?\",\"answer\":\"It partitions the adjacency graph of a planar subdivision into exactly p connected subgraphs to minimize the total dissimilarity across vertex pairs that lie in the same subgraph.\"},{\"question\":\"Why is the p-regions problem hard to solve optimally?\",\"answer\":\"It is NP-hard, and even instances with about 25 areas are reported as challenging for exact methods to reach proven optimality within practical time limits.\"},{\"question\":\"What new ILP models does the paper propose and what do they improve?\",\"answer\":\"The paper presents ER-S, strengthens the existing Tree formulation with new p-regions-specific subtour elimination inequalities, and combines them into ER-S-Tree, 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is the p-regions problem in the paper?","Question",{"text":76,"@type":77},"It partitions the adjacency graph of a planar subdivision into exactly p connected subgraphs to minimize the total dissimilarity across vertex pairs that lie in the same subgraph.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is the p-regions problem hard to solve optimally?",{"text":81,"@type":77},"It is NP-hard, and even instances with about 25 areas are reported as challenging for exact methods to reach proven optimality within practical time limits.",{"name":83,"@type":74,"acceptedAnswer":84},"What new ILP models does the paper propose and what do they improve?",{"text":85,"@type":77},"The paper presents ER-S, strengthens the existing Tree formulation with new p-regions-specific subtour elimination inequalities, and combines them into ER-S-Tree, which dominates prior state-of-the-art models in polyhedral strength and improves experimental 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