Improved Approximation Algorithms for Three-Dimensional Bin Packing

TL;DR

This work advances the theory of 3D geometric packing by delivering a global 6-approximation for 3D-BP, 3D-SP, and 3D-MVBB, and by proving an asymptotic near-2.54-approximation for 3D-BP through harmonic rounding and a 2D-container-packing bridge to 3D configurations. It also establishes an APTAS for 3D-MVBB, and extends the framework to 3D-BP with rotations, achieving a 5-approximation, thereby significantly improving prior absolute and asymptotic guarantees. Central to the approach are volume-based packing lemmas, harmonic transformations, NFDH-based layering, and a Generalized Assignment Problem framing to allocate tall/tall-like items into structured containers. The results bridge 3D packing with 2D container-packings, enabling tighter bounds and practical algorithms with strong theoretical guarantees and potential applicability to higher dimensions and resource-augmented settings.

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 $46/7+\varepsilon$, respectively, for 3D-BP, 3D-SP, and 3D-MVBB. We provide improved approximation ratios of 6, 6, and $3+\varepsilon$, respectively, for the three problems, for any constant $\varepsilon > 0$. 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 $T_{\infty}^2 + \varepsilon\approx 2.86$, where $T_{\infty}$ is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of $3 T_{\infty}/2 +\varepsilon \approx 2.54$. Further, we show that unlike 3D-BP (and 3D-SP), 3D-MVBB admits an APTAS.

Figures (5)

• Figure 1: Packing from \ref{['thm:packseparate']} (only the front view is shown for simplicity) for $k=2$. The dark gray items are sliced while cutting out $\lfloor 3k/2 \rfloor+1$ bins from the Strip Packing solution. Finally, sliced items are packed into the empty regions of the last bin.
• Figure 2: The light gray items are items of $L$. The dark gray items are deleted in order to position the upper large item at a multiple of $\mu^4$.
• Figure 3: The regions between two consecutive dotted lines correspond to slots.
• Figure 4: Packing inside bin $B_h$
• Figure 5: The left figure depicts a 2D container-packing, which forms the base of a 3D configuration. The middle figure shows two 3D containers corresponding to two containers (one big and one horizontal) in 2D container-packing. On the right, the packing that ensures the tall-not-sliced property is shown. The light gray rectangles are repacked into additional bins.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: li-cheng
• Theorem 6: martello2000three
• Theorem 7: 3d-strip-packing
• Lemma 8: coffman1980performance
• Theorem 9: 2dknapsack-lpacking
• Lemma 9