License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.78
URN: urn:nbn:de:0030-drops-164192
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16419/
Jansen, Klaus ;
Khan, Arindam ;
Lira, Marvin ;
Sreenivas, K. V. N.
A PTAS for Packing Hypercubes into a Knapsack
Abstract
We study the d-dimensional hypercube knapsack problem ({d}-D Hc-Knapsack) where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping packing of a subset of hypercubes such that the profit of the packed hypercubes is maximized. For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+ε). For d = 2, Jansen and Solis-Oba (IPCO'08) showed that the problem admits a polynomial-time approximation scheme (PTAS); Heydrich and Wiese (SODA'17) further improved the running time and gave an efficient polynomial-time approximation scheme (EPTAS). Both the results use structural properties of 2-D packing, which do not generalize to higher dimensions. For d > 2, it remains open to obtain a PTAS, and in fact, there has been no improvement since Harren’s result.
We settle the problem by providing a PTAS. Our main technical contribution is a structural lemma which shows that any packing of hypercubes can be converted into another structured packing such that a high profitable subset of hypercubes is packed into a constant number of special hypercuboids, called ?-Boxes and ?-Boxes. As a side result, we give an almost optimal algorithm for a variant of the strip packing problem in higher dimensions. This might have applications for other multidimensional geometric packing problems.
BibTeX - Entry
@InProceedings{jansen_et_al:LIPIcs.ICALP.2022.78,
author = {Jansen, Klaus and Khan, Arindam and Lira, Marvin and Sreenivas, K. V. N.},
title = {{A PTAS for Packing Hypercubes into a Knapsack}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {78:1--78:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16419},
URN = {urn:nbn:de:0030-drops-164192},
doi = {10.4230/LIPIcs.ICALP.2022.78},
annote = {Keywords: Multidimensional knapsack, geometric packing, cube packing, strip packing}
}
|
Keywords: |
|
Multidimensional knapsack, geometric packing, cube packing, strip packing |
|
Collection: |
|
49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
|
Issue Date: |
|
2022 |
|
Date of publication: |
|
28.06.2022 |
