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On the complexity of anchored rectangle packing
Citation Link: https://doi.org/10.15480/882.14814
Publikationstyp
Conference Paper
Date Issued
2019-09
Sprache
English
Author(s)
Cristi, Andrés
Kaaser, Dominik
TORE-DOI
First published in
Number in series
144
Article Number
8
Citation
Leibniz International Proceedings in Informatics, LIPIcs 144 : 8 (2019-09-01)
Contribution to Conference
Publisher DOI
Scopus ID
Publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH, Dagstuhl Publishing
ISBN
978-3-95977-124-5
In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0,1]^2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p in P. In the most prominent variant - Lower-Left-Anchored Rectangle Packing (LLARP) - rectangles are anchored in their lower-left corner. Freedman [W. T. Tutte (Ed.), 1969] conjectured in 1969 that, if (0,0) in P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [Dumitrescu and Tóth, 2015]. Maybe even more surprisingly, it is not known whether LLARP - or any ARP-problem with only one anchor - is NP-hard.
In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies in the interior of the rectangles. Afterwards, we turn to the LLARP problem and investigate two different resource-augmentation settings: In the first we allow an epsilon-perturbation of the input P, whereas in the second we permit an epsilon-overlap between rectangles. For the former setting, we give an algorithm that covers at least as much area as an optimal solution of the original problem. For the latter, we give an (1 - epsilon)-approximation.
In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies in the interior of the rectangles. Afterwards, we turn to the LLARP problem and investigate two different resource-augmentation settings: In the first we allow an epsilon-perturbation of the input P, whereas in the second we permit an epsilon-overlap between rectangles. For the former setting, we give an algorithm that covers at least as much area as an optimal solution of the original problem. For the latter, we give an (1 - epsilon)-approximation.
Subjects
Anchored rectangle
Hardness
NP
PTAS
Rectangle packing
Resource augmentation
DDC Class
600: Technology
Publication version
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