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Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.47
URN: urn:nbn:de:0030-drops-126505
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12650/
Khan, Arindam ;
Pittu, Madhusudhan Reddy
On Guillotine Separability of Squares and Rectangles
Abstract
Guillotine separability of rectangles has recently gained prominence in combinatorial optimization, computational geometry, and combinatorics. Consider a given large stock unit (say glass or wood) and we need to cut out a set of required rectangles from it. Many cutting technologies allow only end-to-end cuts called guillotine cuts. Guillotine cuts occur in stages. Each stage consists of either only vertical cuts or only horizontal cuts. In k-stage packing, the number of cuts to obtain each rectangle from the initial packing is at most k (plus an additional trimming step to separate the rectangle itself from a waste area). Pach and Tardos [Pach and Tardos, 2000] studied the following question: Given a set of n axis-parallel rectangles (in the weighted case, each rectangle has an associated weight), cut out as many rectangles (resp. weight) as possible using a sequence of guillotine cuts. They provide a guillotine cutting sequence that recovers 1/(2 log n)-fraction of rectangles (resp. weights). Abed et al. [Fidaa Abed et al., 2015] claimed that a guillotine cutting sequence can recover a constant fraction for axis-parallel squares. They also conjectured that for any set of rectangles, there exists a sequence of axis-parallel guillotine cuts that recovers a constant fraction of rectangles. This conjecture, if true, would yield a combinatorial O(1)-approximation for Maximum Independent Set of Rectangles (MISR), a long-standing open problem. We show the conjecture is not true, if we only allow o(log log n) stages (resp. o(log n/log log n)-stages for the weighted case). On the positive side, we show a simple O(n log n)-time 2-stage cut sequence that recovers 1/(1+log n)-fraction of rectangles. We improve the extraction of squares by showing that 1/40-fraction (resp. 1/160 in the weighted case) of squares can be recovered using guillotine cuts. We also show O(1)-fraction of rectangles, even in the weighted case, can be recovered for many special cases of rectangles, e.g. fat (bounded width/height), δ-large (large in one of the dimensions), etc. We show that this implies O(1)-factor approximation for Maximum Weighted Independent Set of Rectangles, the weighted version of MISR, for these classes of rectangles.
BibTeX - Entry
@InProceedings{khan_et_al:LIPIcs:2020:12650,
author = {Arindam Khan and Madhusudhan Reddy Pittu},
title = {{On Guillotine Separability of Squares and Rectangles}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages = {47:1--47:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-164-1},
ISSN = {1868-8969},
year = {2020},
volume = {176},
editor = {Jaros{\l}aw Byrka and Raghu Meka},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12650},
URN = {urn:nbn:de:0030-drops-126505},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.47},
annote = {Keywords: Guillotine cuts, Rectangles, Squares, Packing, k-stage packing}
}
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Keywords: |
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Guillotine cuts, Rectangles, Squares, Packing, k-stage packing |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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11.08.2020 |
