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Improved Approximation Algorithms for Three-Dimensional Bin Packing

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Title: Improved Approximation Algorithms for Three-Dimensional Bin Packing
Authors: Kar, Debajyoti, Khan, Arindam, Rau, Malin, 1990
Source: 52nd EATCS International Colloquium on Automata, Languages, and Programming, ICALP 2025, Aarhus, Denmark Leibniz International Proceedings in Informatics, LIPIcs. 334
Subject Terms: Geometric Packing, Approximation Algorithms, Multidimensional Packing
Description: 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 3T∞/2 + ε ≈ 2.54. Further, we show that unlike 3d-bp (and 3d-sp), 3d-mvbb admits an APTAS.
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  Data: Improved Approximation Algorithms for Three-Dimensional Bin Packing
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  Data: <searchLink fieldCode="AR" term="%22Kar%2C+Debajyoti%22">Kar, Debajyoti</searchLink><br /><searchLink fieldCode="AR" term="%22Khan%2C+Arindam%22">Khan, Arindam</searchLink><br /><searchLink fieldCode="AR" term="%22Rau%2C+Malin%22">Rau, Malin</searchLink>, 1990
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  Data: <i>52nd EATCS International Colloquium on Automata, Languages, and Programming, ICALP 2025, Aarhus, Denmark Leibniz International Proceedings in Informatics, LIPIcs</i>. 334
– Name: Subject
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  Data: <searchLink fieldCode="DE" term="%22Geometric+Packing%22">Geometric Packing</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+Algorithms%22">Approximation Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Multidimensional+Packing%22">Multidimensional Packing</searchLink>
– Name: Abstract
  Label: Description
  Group: Ab
  Data: 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 3T∞/2 + ε ≈ 2.54. Further, we show that unlike 3d-bp (and 3d-sp), 3d-mvbb admits an APTAS.
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        Value: 10.4230/LIPIcs.ICALP.2025.104
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      – Text: English
    Subjects:
      – SubjectFull: Geometric Packing
        Type: general
      – SubjectFull: Approximation Algorithms
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      – SubjectFull: Multidimensional Packing
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      – TitleFull: Improved Approximation Algorithms for Three-Dimensional Bin Packing
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            NameFull: Kar, Debajyoti
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            NameFull: Khan, Arindam
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            NameFull: Rau, Malin
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              Y: 2025
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              Value: 334
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            – TitleFull: 52nd EATCS International Colloquium on Automata, Languages, and Programming, ICALP 2025, Aarhus, Denmark Leibniz International Proceedings in Informatics, LIPIcs
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