Non-crossing matchings of points with geometric objects

  • Authors:

  • Greg Aloupis;Jean Cardinal;SéBastien Collette;Erik D. Demaine;Martin L. Demaine;Muriel Dulieu;Ruy Fabila-Monroy;Vi Hart;Ferran Hurtado;Stefan Langerman;Maria Saumell;Carlos Seara;Perouz Taslakian

  • Affiliations:

  • Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium;Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium;Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium;MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA;MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA;Polytechnic Institute of NYU, USA;Departamento de Matemáticas, CINVESTAV, México DF, Mexico;Stony Brook University, Stony Brook, NY 11794, USA;Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain;Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium;Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain;Universitat Politècnica de Catalunya, Jordi Girona 1-3, E-08034 Barcelona, Spain;Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2013

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Abstract

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point-object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.