On Approximation Schemes for Stabbing Rectilinear Polygons

We study the problem of stabbing rectilinear polygons, where we are given n rectilinear polygons in the plane that we want to stab, i.e., we want to select horizontal line segments such that for each given rectilinear polygon there is a line segment that intersects two opposite (parallel) edges of it. Our goal is to find a set of line segments of minimum total length such that all polygons are stabbed. For the special case of rectangles, there is an O(1)-approximation algorithm and the problem is NP-hard [Chan, van Dijk, Fleszar, Spoerhase, and Wolff, 2018]. Also, the problem admits a QPTAS [Eisenbrand, Gallato, Svensson, and Venzin, 2021] and even a PTAS [Khan, Subramanian, and Wiese, 2022]. However, the approximability for the setting of more general polygons, e.g., L-shapes or T-shapes, is completely open. In this paper, we give conditions under which the problem admits a (1 + å)-approximation algorithm. We assume that each input polygon is composed of rectangles that are placed on top of each other. We show that if all input polygons satisfy the hourglass condition, then the problem admits a quasi-polynomial time approximation scheme. In particular, it is thus unlikely that this case is APX-hard. Furthermore, we show that there exists a PTAS if each input polygon is composed out of rectangles with a bounded range of widths. On the other hand, we prove that the general case of the problem (in which the input polygons may not satisfy these conditions) is APX-hard, already if all input polygons have only eight edges. We remark that all polygons with fewer edges automatically satisfy the hourglass condition. For arbitrary rectilinear polygons we even show a lower bound of Ω(log n) for the possible approximation ratio, which implies that the best possible ratio is in Θ(log n) since the problem is a special case of Set Cover.