Minimum Perimeter-Sum Partitions in the Plane

  • Mikkel AbrahamsenEmail author
  • Mark de Berg
  • Kevin Buchin
  • Mehran Mehr
  • Ali D. Mehrabi


Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets \(P_1\) and \(P_2\) such that the sum of the perimeters of \({\textsc {ch}}(P_1)\) and \({\textsc {ch}}(P_2)\) is minimized, where \({\textsc {ch}}(P_i)\) denotes the convex hull of \(P_i\). The problem was first studied by Mitchell and Wynters in 1991 who gave an \(O(n^2)\) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in \(O(n \log ^2 n)\) time and a \((1+\varepsilon )\)-approximation algorithm running in \(O(n + 1/\varepsilon ^2\cdot \log ^2(1/\varepsilon ))\) time.


Computational geometry Clustering Minimum-perimeter partition Convex hull 

Mathematics Subject Classification




This research was initiated when the first author visited the Department of Computer Science at TU Eindhoven during the winter 2015–2016. He wishes to express his gratitude to the other authors and the department for their hospitality.


M. Abrahamsen is supported by the Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research under the Sapere Aude research career programme. M. de Berg, K. Buchin, M. Mehr, and A. D. Mehrabi are supported by the Netherlands’ Organisation for Scientific Research (NWO) under Project No. 024.002.003, 612.001.207, 022.005025, and 612.001.118 respectively.


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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands

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