Geometric Multicut

Authors Mikkel Abrahamsen , Panos Giannopoulos, Maarten Löffler, Günter Rote

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Author Details

Mikkel Abrahamsen
  • BARC, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark
Panos Giannopoulos
  • giCenter, Department of Computer Science, City University of London, EC1V 0HB, London, UK
Maarten Löffler
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Günter Rote
  • Institut für Informatik, Freie Universität Berlin, Takustraße 9, 14195 Berlin, Germany


This work was initiated at the workshop on Fixed-Parameter Computational Geometry at the Lorentz Center in Leiden in May 2018. We thank the organizers and the Lorentz Center for a nice workshop and Michael Hoffmann for useful discussions during the workshop.

Cite AsGet BibTex

Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, and Günter Rote. Geometric Multicut. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" F, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, where k is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an O(n^4 log^3 n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colors and n corners in total. We then show that the problem is NP-hard for the case of three colors. Finally, we give a (2-4/3k)-approximation algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
  • multicut
  • clustering
  • Steiner tree


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