New results on stabbing segments with a polygon

J. M. Díaz-Báñez, M. Korman, P. Pérez-Lantero, A. Pilz, C. Seara, and R. I. Silveira

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)] [16] 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.

Reference: J. M. Díaz-Báñez, M. Korman, P. Pérez-Lantero, A. Pilz, C. Seara, and R. I. Silveira. New results on stabbing segments with a polygon. Comput. Geom., 48(1):14-29, 2015.