A PTAS for the Horizontal Rectangle Stabbing Problem

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 [5] initiated the study of these problems by providing O(1)-approximation algorithms. Recently, Eisenbrand, Gallato, Svensson, and Venzin [11] have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing but it is 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.