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

Timothy M. Chan, Thomas C. van Dijk, Krzysztof Fleszar, Joachim Spoerhase, Alexander Wolff

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Abstract

We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have
been considered so far. Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is (log n), it is an important field in geometric
approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA’12] generalize earlier results by Varadarajan [STOC’10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity. Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.
Original languageEnglish
Title of host publication29th International Symposium on Algorithms and Computation (ISAAC 2018)
Pages1-13
ISBN (Electronic)978-3-95977-094-1
DOIs
Publication statusPublished - 2018
MoE publication typeA4 Article in a conference publication
EventInternational Symposium on Algorithms and Computation - Jiaoxi, Taiwan, Republic of China
Duration: 16 Dec 201819 Dec 2018
Conference number: 29

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum für Informatik
Volume123
ISSN (Electronic)1868-8969

Conference

ConferenceInternational Symposium on Algorithms and Computation
Abbreviated titleISAAC
CountryTaiwan, Republic of China
CityJiaoxi
Period16/12/201819/12/2018

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