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Crossing and Independent Families Among Polygons

Authors Anna Brötzner , Robert Ganian , Thekla Hamm , Fabian Klute , Irene Parada

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • crossing families
  • crossing-free matchings
  • segment intersection graphs
  • computational geometry
  • parameterized algorithms

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Abstract

Given a set A of points in the plane, a family of line segments forming a matching in A is called crossing (or independent) if each pair of segments in the family intersects (or is non-intersecting, respectively). In past works, these notions have been generalized to polygons by identifying the points in A with the vertices of a given set of polygons and forbidding the line segments from intersecting or overlapping with polygon walls. In this work, we study the computational complexity of computing maximum crossing and independent families in this more general setting.
As our first two results, we show that both problems are NP-hard already when the polygons are triangles. Motivated by this, we turn to parameterized algorithms. For our main algorithmic results, we consider the number of polygons on the input as the natural parameter and under this parameterization obtain a fixed-parameter algorithm for computing a largest crossing family among these polygons, and a separate XP-algorithm for computing a largest independent family that lies in one of the faces of the polygonal domain.

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Anna Brötzner, Robert Ganian, Thekla Hamm, Fabian Klute, and Irene Parada. Crossing and Independent Families Among Polygons. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.11

Author Details

Anna Brötzner
  • Malmö University, Sweden
Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Thekla Hamm
  • TU Eindhoven, The Netherlands
Fabian Klute
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Irene Parada
  • Universitat Politècnica de Catalunya, Barcelona, Spain

Funding

  • Brötzner, Anna: supported by grant 2021-03810 (Illuminate: provably good algorithms for guarding problems) from the Swedish Research Council (Vetenskapsrådet).
  • Ganian, Robert: Projects No. 10.55776/Y1329 and 10.55776/COE12 of the Austrian Science Fund (FWF), Project No. 10.47379/ICT22029 of the Vienna Science Foundation (WWTF).
  • Klute, Fabian: Partially supported by grant PID2023-150725NB-I00 funded by MICIU/AEI/ 10.13039/501100011033.
  • Parada, Irene: Serra Húnter Fellow. Partially supported by grant PID2023-150725NB-I00 funded by MICIU/AEI/10.13039/501100011033.

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