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A PTAS for Packing Hypercubes into a Knapsack

Authors Klaus Jansen, Arindam Khan , Marvin Lira, K. V. N. Sreenivas

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Multidimensional knapsack
  • geometric packing
  • cube packing
  • strip packing

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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 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.

Cite As Get BibTex

Klaus Jansen, Arindam Khan, Marvin Lira, and K. V. N. Sreenivas. A PTAS for Packing Hypercubes into a Knapsack. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.78

Author Details

Klaus Jansen
  • Universität Kiel, Germany
Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Marvin Lira
  • Universität Kiel, Germany
K. V. N. Sreenivas
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

Funding

  • Jansen, Klaus: Research supported by German Research Foundation (DFG), project JA 612/25-1.
  • Khan, Arindam: Research partly supported by Pratiksha Trust Young Investigator Award, Google India Research Award, and Google ExploreCS Award.
  • Lira, Marvin: Research supported by German Research Foundation (DFG), project JA 612/25-1.

Acknowledgements

We thank Roberto Solis-Oba and three anonymous reviewers for their comments and suggestions on this paper.

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