Improved Approximation Algorithms for Three-Dimensional Bin Packing

Paper i proceeding, 2025

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.