License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ICALP.2022.78
URN: urn:nbn:de:0030-drops-164192
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Jansen, Klaus ; Khan, Arindam ; Lira, Marvin ; Sreenivas, K. V. N.

A PTAS for Packing Hypercubes into a Knapsack

LIPIcs-ICALP-2022-78.pdf (0.8 MB)


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 𝒱-Boxes and 𝒩-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

  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 =		{},
  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

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