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A Parameterized Approximation Scheme for the Geometric Knapsack Problem with Wide Items

Authors Mathieu Mari, Timothé Picavet , Michał Pilipczuk



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Mathieu Mari
  • Institute of Informatics, University of Warsaw, Poland
  • IDEAS-NCBR, Warsaw, Poland
Timothé Picavet
  • ENS de Lyon, France
  • Aalto University, Finland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland

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Mathieu Mari, Timothé Picavet, and Michał Pilipczuk. A Parameterized Approximation Scheme for the Geometric Knapsack Problem with Wide Items. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 33:1-33:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.33

Abstract

We study a natural geometric variant of the classic Knapsack problem called 2D-Knapsack: we are given a set of axis-parallel rectangles and a rectangular bounding box, and the goal is to pack as many of these rectangles inside the box without overlap. Naturally, this problem is NP-complete. Recently, Grandoni et al. [ESA'19] showed that it is also 𝖶[1]-hard when parameterized by the size k of the sought packing, and they presented a parameterized approximation scheme (PAS) for the variant where we are allowed to rotate the rectangles by 90° before packing them into the box. Obtaining a PAS for the original 2D-Knapsack problem, without rotation, appears to be a challenging open question. In this work, we make progress towards this goal by showing a PAS under the following assumptions: - both the box and all the input rectangles have integral, polynomially bounded sidelengths; - every input rectangle is wide - its width is greater than its height; and - the aspect ratio of the box is bounded by a constant. Our approximation scheme relies on a mix of various parameterized and approximation techniques, including color coding, rounding, and searching for a structured near-optimum packing using dynamic programming.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Packing and covering problems
Keywords
  • Parameterized complexity
  • Approximation scheme
  • Geometric knapsack
  • Color coding
  • Dynamic programming
  • Computational geometry

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References

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