Non-crossing matchings of points with geometric objects

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 pointobject 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 number is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and Email addresses: [email protected] (Greg Aloupis), [email protected] (Jean Cardinal), [email protected] (Sébastien Collette), [email protected] (Erik D. Demaine), [email protected] (Martin L. Demaine), [email protected] (Muriel Dulieu), [email protected] (Ruy Fabila-Monroy), [email protected] (Vi Hart), [email protected] (Ferran Hurtado), [email protected] (Stefan Langerman), [email protected] (Maria Saumell), [email protected] (Carlos Seara), [email protected] (Perouz Taslakian) Supported by the Communauté française de Belgique ARC. Chargé de recherches du F.R.S.-FNRS. Mâıtre de recherches du F.R.S.-FNRS. Partially supported by projects MTM2009-07242 and Gen. Cat. DGR 2009SGR1040. Preprint submitted to Elsevier September 2, 2010 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.