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 a $O(1)$-approximation algorithm and the problem is $\mathsf{NP}$-hard [Chan et al.]. Also, the problem admits a QPTAS [Eisenbrand et al.] and even a PTAS [Khan et al.]. However, the approximability for the setting of more general polygons, e.g., L-shapes or T-shapes, is completely open.
In this paper, we characterize the conditions under which the problem admits a $(1+\varepsilon)$-approximation algorithm. We assume that each input polygon is composed of rectangles that are placed on top of each other such that, for each pair of adjacent edges between rectangles, one edge contains the other. We show that if all input polygons satisfy the hourglass condition, then the problem admits a QPTAS. In particular, it is thus unlikely that this case is $\mathsf{APX}$-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, if the input polygons do not satisfy these conditions, we prove that the problem is $\mathsf{APX}$-hard, already if all input polygons have only eight edges. We remark that all polygons with fewer edges automatically satisfy the hourglass condition. On the other hand, 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.
