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On Approximation Schemes for Stabbing Rectilinear Polygons

Arindam Khan, Aditya Subramanian, Tobias Widmann, Andreas Wiese

TL;DR

The paper investigates stabbing rectilinear polygons with minimum total length of horizontal segments, extending from rectangles to $k$-shapes. It establishes a precise boundary for tractability: a $(1+\varepsilon)$-approximation is achievable in quasi-polynomial time under the hourglass condition (and a PTAS under bounded width ranges), while violated cases yield $\mathsf{APX}$-hardness, with a universal $\Theta(\log n)$-inapproximability for general rectilinear polygons. The core method for the hourglass case is a hierarchical plane partition that guesses long segments and recurses on subregions, leveraging the connectedness preserved by the hourglass property. Collectively, the results delineate when high-accuracy approximations are feasible for stabbing $k$-shapes and link geometric stabbing to classic set cover hardness, shaping both theory and potential applications in resource allocation and geometric optimization.

Abstract

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.

On Approximation Schemes for Stabbing Rectilinear Polygons

TL;DR

The paper investigates stabbing rectilinear polygons with minimum total length of horizontal segments, extending from rectangles to -shapes. It establishes a precise boundary for tractability: a -approximation is achievable in quasi-polynomial time under the hourglass condition (and a PTAS under bounded width ranges), while violated cases yield -hardness, with a universal -inapproximability for general rectilinear polygons. The core method for the hourglass case is a hierarchical plane partition that guesses long segments and recurses on subregions, leveraging the connectedness preserved by the hourglass property. Collectively, the results delineate when high-accuracy approximations are feasible for stabbing -shapes and link geometric stabbing to classic set cover hardness, shaping both theory and potential applications in resource allocation and geometric optimization.

Abstract

We study the problem of stabbing rectilinear polygons, where we are given 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 -approximation algorithm and the problem is -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 -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 -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 -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 for the possible approximation ratio, which implies that the best possible ratio is in since the problem is a special case of Set Cover.
Paper Structure (5 sections, 2 theorems, 3 figures)

This paper contains 5 sections, 2 theorems, 3 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Other related work
  4. Preliminaries
  5. Quasi-polynomial-time approximation scheme

Key Result

Lemma 3

By losing a factor of $1+\varepsilon$ in our approximation ratio, we assume that $\frac{\varepsilon}{n}<w_{\operatorfont min}(\mathcal{K})\leq w_{\operatorfont max}(\mathcal{K})\leq\log n$ and $w_{\operatorfont range}(\mathcal{K})\le n\log n$.

Figures (3)

  • Figure 1: Examples of $k$-shapes satisfying the hourglass condition
  • Figure 2: A 3-shape not satisfying the hourglass condition (left), and a stack of rectangles that does not form a $k$-shape (right).
  • Figure 3: Partitioning the instance into narrow strips.

Theorems & Definitions (5)

  • Definition 1: $k$-shape
  • Definition 2
  • Definition 3
  • Lemma 3
  • Lemma 3: Partitioning into narrow strips