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Improved Approximation Algorithms for Three-Dimensional Bin Packing

Authors Debajyoti Kar , Arindam Khan , Malin Rau

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ACM Subject Classification
  • Theory of computation → Packing and covering problems
Keywords
  • Approximation Algorithms
  • Geometric Packing
  • Multidimensional Packing

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Abstract

We study three fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), 3D Strip Packing (3D-SP), and Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal is to find an axis-aligned nonoverlapping packing of all cuboids. In 3D-BP, we need to pack the given cuboids into the minimum number of unit cube bins. In 3D-SP, we need to pack them into a 3D cuboid with a unit square base and minimum height. Finally, in 3D-MVBB, the goal is to pack into a cuboid box of minimum volume. 
It is NP-hard to even decide whether a set of rectangles can be packed into a unit square bin - giving an (absolute) approximation hardness of 2 for 3D-BP and 3D-SP. The previous best (absolute) approximation for all three problems is by Li and Cheng (SICOMP, 1990), who gave algorithms with approximation ratios of 13, 46/7, and 46/7+ε, respectively, for 3D-BP, 3D-SP, and 3D-MVBB. We provide improved approximation ratios of 6, 6, and 3+ε, respectively, for the three problems, for any constant ε > 0.
For 3D-BP, in the asymptotic regime, Bansal, Correa, Kenyon, and Sviridenko (Math. Oper. Res., 2006) showed that there is no asymptotic polynomial-time approximation scheme (APTAS) even when all items have the same height. Caprara (Math. Oper. Res., 2008) gave an asymptotic approximation ratio of T_{∞}² + ε ≈ 2.86, where T_{∞} is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of 3 T_{∞}/2 + ε ≈ 2.54. Further, we show that unlike 3D-BP (and 3D-SP), 3D-MVBB admits an APTAS.

Cite As Get BibTex

Debajyoti Kar, Arindam Khan, and Malin Rau. Improved Approximation Algorithms for Three-Dimensional Bin Packing. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 104:1-104:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.104

Author Details

Debajyoti Kar
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Malin Rau
  • Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden

Funding

  • Kar, Debajyoti: Google PhD Fellowship.
  • Khan, Arindam: Google India Research Award, SERB Core Research Grant (CRG/2022/001176) on "Optimization under Intractability and Uncertainty", Ittiam Systems CSR grant, and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).

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