New results on stabbing segments with a polygon

Abstract

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236–269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard.

DOI: 10.1016/j.comgeo.2014.06.002

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Cite this paper

@article{DazBez2013NewRO, title={New results on stabbing segments with a polygon}, author={Jos{\'e} Miguel D{\'i}az-B{\'a}{\~n}ez and Matias Korman and Pablo P{\'e}rez-Lantero and Alexander Pilz and Carlos Seara and Rodrigo I. Silveira}, journal={Comput. Geom.}, year={2013}, volume={48}, pages={14-29} }