New results on stabbing segments with a polygon

TL;DR

This paper investigates the problem of finding polygons that stab segments by containing at least one endpoint, providing polynomial solutions for disjoint segments and proving NP-hardness for general segments, also addressing an open problem on convex hulls of imprecise points.

Contribution

It introduces a polynomial-time algorithm for stabbing disjoint segments with a polygon and proves NP-hardness for the general case, also solving an open problem on convex hulls of imprecise points.

Findings

Polynomial-time algorithm for disjoint segments

NP-hardness for general segments

Solution to an open problem on convex hulls of imprecise points

Abstract

We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon $\mathcal{P}$ if at least one of its two endpoints is contained in $\mathcal{P}$. A segment set $S$ is stabbed by $\mathcal{P}$ if every segment of $S$ is stabbed by $\mathcal{P}$. We show that if $S$ is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing $S$ can be solved in polynomial time. We also prove that for general segments the problem is NP-hard. Further, an adaptation of our polynomial-time algorithm solves an open problem posed by L\"offler 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.