A PTAS for the horizontal rectangle stabbing problem

TL;DR

This paper presents a Polynomial-Time Approximation Scheme (PTAS) for the horizontal rectangle stabbing problem, advancing the understanding of approximation algorithms for geometric stabbing problems.

Contribution

It provides the first PTAS for the horizontal rectangle stabbing problem and improved approximation algorithms for the general rectangle stabbing problem and its special cases.

Findings

Established a PTAS for the horizontal rectangle stabbing problem.

Achieved a (2+ε)-approximation for the general rectangle stabbing problem.

Developed PTASes for special cases like squares, width ≤ height, and δ-large rectangles.

Abstract

We study rectangle stabbing problems in which we are given $n$ axis-aligned rectangles in the plane that we want to stab, i.e., we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (STABBING), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In general rectangle stabbing problem, also known as horizontal-vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (i.e., either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan, van Dijk, Fleszar, Spoerhase, and Wolff [2018]initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand, Gallato, Svensson, and Venzin [2021] have presented a QPTAS and a polynomial-time 8-approximation algorithm for STABBING but it is was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for STABBING, settling this question. For HV-Stabbing, we obtain a $(2+\varepsilon)$-approximation. We also obtain PTASes for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are $\delta$-large, i.e., have at least one edge whose length is at least $\delta$, while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.