Document Open Access Logo

Bicriteria Polygon Aggregation with Arbitrary Shapes

Authors Lotte Blank , David Eppstein, Jan-Henrik Haunert , Herman Haverkort, Benedikt Kolbe , Philip Mayer , Petra Mutzel , Alexander Naumann , Jonas Sauer

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.11.pdf
  • Filesize: 3.66 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Information systems → Geographic information systems
Keywords
  • polygon aggregation
  • fencing
  • minimum-perimeter clustering
  • map generalization
  • urban analytics

Metrics

Abstract

We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime Õ(n⁴), where n is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that 𝒪 combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.

Cite As Get BibTex

Lotte Blank, David Eppstein, Jan-Henrik Haunert, Herman Haverkort, Benedikt Kolbe, Philip Mayer, Petra Mutzel, Alexander Naumann, and Jonas Sauer. Bicriteria Polygon Aggregation with Arbitrary Shapes. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.11

Author Details

Lotte Blank
  • University of Bonn, Germany
David Eppstein
  • University of California, Irvine, CA, USA
Jan-Henrik Haunert
  • University of Bonn, Germany
Herman Haverkort
  • University of Bonn, Germany
Benedikt Kolbe
  • University of Bonn, Germany
  • Hausdorff Center for Mathematics, Bonn, Germany
  • Lamarr Institute, Bonn, Germany
Philip Mayer
  • University of Bonn, Germany
Petra Mutzel
  • University of Bonn, Germany
  • Lamarr Institute, Bonn, Germany
Alexander Naumann
  • University of Bonn, Germany
Jonas Sauer
  • Karlsruhe Institute of Technology, Germany

Funding

This research was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant FOR-5361 - 459420781.
  • Eppstein, David: Research supported in part by NSF grant CCF-2212129.
  • Kolbe, Benedikt: This research has partly been funded by the Federal Ministry of Education and Research of Germany and the state of North-Rhine Westphalia as part of the Lamarr-Institute for Machine Learning and Artificial Intelligence.

Acknowledgements

We thank Anne Driemel and Frederik Brüning for fruitful initial discussions. Additionally, we thank the anonymous reviewers for helping to improve this paper.

Supplementary Materials

References

  1. Mikkel Abrahamsen, Anna Adamaszek, Karl Bringmann, Vincent Cohen-Addad, Mehran Mehr, Eva Rotenberg, Alan Roytman, and Mikkel Thorup. Fast fencing. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 564-573. Association for Computing Machinery, 2018. URL: https://doi.org/10.1145/3188745.3188878.
  2. Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, and Günter Rote. Geometric multicut: Shortest fences for separating groups of objects in the plane. Discrete & Computational Geometry, 64(3):575-607, 2020. URL: https://doi.org/10.1007/S00454-020-00232-W.
  3. Karl Heinrich Anders and Monika Sester. Parameter-free cluster detection in spatial databases and its application to typification. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 33:75-82, 2000. URL: https://www.isprs.org/proceedings/xxxiii/congress/part4/75_XXXIII-part4.pdf.
  4. Esther M. Arkin, Samir Khuller, and Joseph S. B. Mitchell. Geometric knapsack problems. In Algorithms and Data Structures, 2nd Workshop WADS '91, volume 519 of Lecture Notes in Computer Science, pages 165-176. Springer, 1991. URL: https://doi.org/10.1007/BFB0028259.
  5. Daniel Arribas-Bel, M.-À. Garcia-López, and Elisabet Viladecans-Marsal. Building(s and) cities: Delineating urban areas with a machine learning algorithm. Journal of Urban Economics, 125:103217, 2021. Delineation of Urban Areas. URL: https://doi.org/10.1016/j.jue.2019.103217.
  6. Arne Beines, Michael Kaibel, Philip Mayer, Petra Mutzel, and Jonas Sauer. A simpler approach for monotone parametric minimum cut: Finding the breakpoints in order. In Proceedings of the 27th Workshop on Algorithm Engineering and Experiments (ALENEX'25), pages 29-41. Society for Industrial and Applied Mathematics (SIAM), 2025. URL: https://doi.org/10.1137/1.9781611978339.3.
  7. Annika Bonerath, Benjamin Niedermann, and Jan-Henrik Haunert. Retrieving α-shapes and schematic polygonal approximations for sets of points within queried temporal ranges. In Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, SIGSPATIAL 2019, pages 249-258. Association for Computing Machinery, 2019. URL: https://doi.org/10.1145/3347146.3359087.
  8. Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum, and Christian Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. SIAM Journal on Computing, 46(4):1280-1303, 2017. URL: https://doi.org/10.1137/15M1042929.
  9. Yuri Boykov and Vladimir Kolmogorov. An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(9):1124-1137, 2004. URL: https://doi.org/10.1109/TPAMI.2004.60.
  10. Kevin Buchin, Wouter Meulemans, André Van Renssen, and Bettina Speckmann. Area-preserving simplification and schematization of polygonal subdivisions. ACM Transactions on Spatial Algorithms and Systems, 2(1), 2016. URL: https://doi.org/10.1145/2818373.
  11. Kevin Buchin, Wouter Meulemans, and Bettina Speckmann. A new method for subdivision simplification with applications to urban-area generalization. In Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, GIS '11, pages 261-270. Association for Computing Machinery, 2011. URL: https://doi.org/10.1145/2093973.2094009.
  12. Vasilis Capoyleas, Günter Rote, and Gerhard J. Woeginger. Geometric clusterings. J. Algorithms, 12(2):341-356, 1991. URL: https://doi.org/10.1016/0196-6774(91)90007-L.
  13. Boost Contributors. Boost C++ Libraries. Boost.org, 2024. Version 1.74. URL: https://www.boost.org/.
  14. Jonathan Damen, Marc van Kreveld, and Bert Spaan. High quality building generalization by extending the morphological operators. In Proceedings of the 11th ICA Workshop on Generalisation and Multiple Representation, pages 1-12, 2008. URL: https://kartographie.geo.tu-dresden.de/downloads/ica-gen/workshop2008/04_Damen_et_al.pdf.
  15. Marie-Pierre de Bellefon, Pierre-Philippe Combes, Gilles Duranton, Laurent Gobillon, and Clément Gorin. Delineating urban areas using building density. Journal of Urban Economics, 125:103226, 2021. URL: https://doi.org/10.1016/j.jue.2019.103226.
  16. Matt Duckham, Lars Kulik, Mike Worboys, and Antony Galton. Efficient generation of simple polygons for characterizing the shape of a set of points in the plane. Pattern Recognition, 41(10):3224-3236, 2008. URL: https://doi.org/10.1016/j.patcog.2008.03.023.
  17. Herbert Edelsbrunner and Ernst P. Mücke. Three-dimensional alpha shapes. ACM Transactions on Graphics, 13(1):43-72, 1994. URL: https://doi.org/10.1145/174462.156635.
  18. Stefan Funke and Sabine Storandt. Smooth building footprint aggregation with alpha shapes. In Web and Wireless Geographical Information Systems: 21st International Symposium, W2GIS 2024, Proceedings, number 14673 in Lecture Notes in Computer Science, pages 117-129. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-60796-7_9.
  19. Pawel Gawrychowski and Adam Karczmarz. Improved bounds for shortest paths in dense distance graphs. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, volume 107 of LIPIcs, pages 61:1-61:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.61.
  20. Oliver Harig, Robert Hecht, Dirk Burghardt, and Gotthard Meinel. Automatic delineation of urban growth boundaries based on topographic data using Germany as a case study. ISPRS International Journal of Geo-Information, 10(5), 2021. URL: https://doi.org/10.3390/ijgi10050353.
  21. Lars Harrie, Hanna Stigmar, and Milan Djordjevic. Analytical estimation of map readability. ISPRS International Journal of Geo-Information, 4(2):418-446, 2015. URL: https://doi.org/10.3390/ijgi4020418.
  22. Jan-Henrik Haunert and Alexander Wolff. Optimal and topologically safe simplification of building footprints. In Proceedings of the 18th SIGSPATIAL International Conference on Advances in Geographic Information Systems, GIS '10, pages 192-201. Association for Computing Machinery, 2010. URL: https://doi.org/10.1145/1869790.1869819.
  23. Adolf Hurwitz. Sur le probleme des isopérimètres. Comptes rendus de l'Académie des Sciences, 132:401-403, 1901. Google Scholar
  24. Safe Software Inc. FME AreaAmalgamator Transformer, 2025. Accessed: 2025-03-24. URL: https://docs.safe.com/fme/html/FME-Form-Documentation/FME-Transformers/Transformers/areaamalgamator.htm.
  25. Warren C. Jochem, Douglas R. Leasure, Oliver Pannell, Heather R. Chamberlain, Patricia Jones, and Andrew J. Tatem. Classifying settlement types from multi-scale spatial patterns of building footprints. Environment and Planning B: Urban Analytics and City Science, 48(5):1161-1179, 2021. URL: https://doi.org/10.1177/2399808320921208.
  26. Alan M. MacEachren. Map complexity: Comparison and measurement. The American Cartographer, 9(1):31-46, 1982. URL: https://doi.org/10.1559/152304082783948286.
  27. Robert B. McMaster and K. Stuart Shea. Generalization in Digital Cartography. Association of American Geographers, 1992. Google Scholar
  28. Adriano J. C. Moreira and Maribel Yasmina Santos. Concave hull: A k-nearest neighbours approach for the computation of the region occupied by a set of points. In GRAPP 2007, Proceedings of the Second International Conference on Computer Graphics Theory and Applications, pages 61-68. INSTICC - Institute for Systems and Technologies of Information, Control and Communication, 2007. URL: https://doi.org/10.5220/0002080800610068.
  29. OpenStreetMap Contributors. OpenStreetMap. https://www.openstreetmap.org, 2004-present.
  30. Dongliang Peng and Guillaume Touya. Continuously generalizing buildings to built-up areas by aggregating and growing. In Proceedings of the 3rd ACM SIGSPATIAL Workshop on Smart Cities and Urban Analytics, UrbanGIS'17. Association for Computing Machinery, 2017. URL: https://doi.org/10.1145/3152178.3152188.
  31. Peter Rottmann, Anne Driemel, Herman Haverkort, Heiko Röglin, and Jan-Henrik Haunert. Bicriteria shapes: Hierarchical grouping and aggregation of polygons with an efficient graph-cut approach. ACM Transactions on Spatial Algorithms and Systems, 11(1), 2025. URL: https://doi.org/10.1145/3705001.
  32. Jacob Steiner. Einfache Beweise der isoperimetrischen Hauptsätze. Journal für die reine und angewandte Mathematik, 18:281-296, 1838. URL: http://eudml.org/doc/147053.
  33. H. Taubenböck, H. Debray, C. Qiu, M. Schmitt, Y. Wang, and X.X. Zhu. Seven city types representing morphologic configurations of cities across the globe. Cities, 105:102814, 2020. URL: https://doi.org/10.1016/j.cities.2020.102814.
  34. The CGAL Project. CGAL User and Reference Manual, 5.6 edition, 2023. URL: https://doc.cgal.org/5.6/Manual/packages.html.
  35. Guillaume Touya and Marion Dumont. Progressive Block Graying and Landmarks Enhancing as Intermediate Representations between Buildings and Urban Areas. In Proceedings of 20th ICA Workshop on Generalisation and Multiple Representation, July 2017. URL: https://hal.science/hal-02097486.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact