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A Tight \((3/2+\varepsilon )\)-Approximation for Skewed Strip Packing

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Abstract

In the Strip Packing problem, we are given a vertical half-strip \([0,W]\times [0,+\infty )\) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. It is NP-hard to approximate this problem within a factor \((3/2-\varepsilon )\) for any constant \(\varepsilon >0\) by a simple reduction from the Partition problem, while the current best approximation factor for it is \((5/3+\varepsilon )\). It seems plausible that Strip Packing admits a \((3/2+\varepsilon )\)-approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter \(\delta >0\), we call large the rectangles with width at least \(\delta W\) and height at least \(\delta OPT\), and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, the skewed case is still NP-hard to approximate within a factor \((3/2-\varepsilon )\), and we provide an (almost) tight \((3/2+\varepsilon )\)-approximation algorithm.

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Notes

  1. Choosing \(\delta \) such that \(\delta W<1\) enforces each rectangle to have height at most \(\delta OPT\) (otherwise it would be large). A PTAS for this case follows, e.g., from [25].

  2. The same item might have different profit on different bins; however, we do not need this extension here.

  3. A shifting up by \(2\varepsilon OPT\) would be sufficient to achieve a contradiction here. The extra shift by \(9\varepsilon OPT\) is used to create some more free space that is needed in the following arguments.

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Authors and Affiliations

  1. Universidad de O’Higgins, Rancagua, Chile

    Waldo Gálvez

  2. IDSIA, USI-SUPSI, Lugano, Switzerland

    Fabrizio Grandoni

  3. Eindhoven University of Technology, Eindhoven, Netherlands

    Afrouz Jabal Ameli

  4. University of Kiel, Kiel, Germany

    Klaus Jansen

  5. Indian Institute of Science, Bengaluru, India

    Arindam Khan

  6. Universität Hamburg, Hamburg, Germany

    Malin Rau

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  1. Waldo Gálvez
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Correspondence to Waldo Gálvez.

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A preliminary version of this paper appeared in the 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2020).

Waldo Gálvez is supported by ANID via Subvención a la Instalación Académica, Folio 85220118. Fabrizio Grandoni and Afrouz Jabal Ameli are partially supported by the SNSF Excellence Grant 200020B_182865/1.

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Gálvez, W., Grandoni, F., Ameli, A.J. et al. A Tight \((3/2+\varepsilon )\)-Approximation for Skewed Strip Packing. Algorithmica 85, 3088–3109 (2023). https://doi.org/10.1007/s00453-023-01130-2

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