A PTAS for Packing Hypercubes into a Knapsack

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

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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


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

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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)


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.

Subject Classification

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


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