Convex partial transversals of planar regions

TL;DR

This paper studies the computational problem of finding convex shapes intersecting a subset of planar regions, providing polynomial algorithms for some cases and NP-hardness results for others, highlighting complexity boundaries.

Contribution

It introduces a polynomial time algorithm for convex partial transversals of disjoint line segments with limited orientations and proves NP-hardness for intersecting rectangles and 3-oriented segments.

Findings

Polynomial algorithm for disjoint segments with limited orientations

NP-hardness for intersecting axis-aligned rectangles

NP-hardness for 3-oriented line segments

Abstract

We consider the problem of testing, for a given set of planar regions $\cal R$ and an integer $k$, whether there exists a convex shape whose boundary intersects at least $k$ regions of $\cal R$. We provide a polynomial time algorithm for the case where the regions are disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.