A QPTAS for stabbing rectangles

07/14/2021
by   Friedrich Eisenbrand, et al.
0

We consider the following geometric optimization problem: Given n axis-aligned rectangles in the plane, the goal is to find a set of horizontal segments of minimum total length such that each rectangle is stabbed. A segment stabs a rectangle if it intersects both its left and right edge. As such, this stabbing problem falls into the category of weighted geometric set cover problems for which techniques that improve upon the general Θ(log n)-approximation guarantee have received a lot of attention in the literature. Chan at al. (2018) have shown that rectangle stabbing is NP-hard and that it admits a constant-factor approximation algorithm based on Varadarajan's quasi-uniform sampling method. In this work we make progress on rectangle stabbing on two fronts. First, we present a quasi-polynomial time approximation scheme (QPTAS) for rectangle stabbing. Furthermore, we provide a simple 8-approximation algorithm that avoids the framework of Varadarajan. This settles two open problems raised by Chan et al. (2018).

READ FULL TEXT

page 1

page 2

page 3

page 4

06/07/2018

Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

We initiate the study of the following natural geometric optimization pr...
11/09/2021

A PTAS for the horizontal rectangle stabbing problem

We study rectangle stabbing problems in which we are given n axis-aligne...
06/26/2019

Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

The area of parameterized approximation seeks to combine approximation a...
02/06/2021

The Maximum Exposure Problem

Given a set of points P and axis-aligned rectangles ℛ in the plane, a po...
09/19/2019

Structured Discrete Shape Approximation: Theoretical Complexity and Practical Algorithm

We consider the problem of approximating a two-dimensional shape contour...
06/01/2021

A (2+ε)-Approximation Algorithm for Maximum Independent Set of Rectangles

We study the Maximum Independent Set of Rectangles (MISR) problem, where...
11/29/2020

A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane

Given two points s and t in the plane and a set of obstacles defined by ...