Improved Approximation Algorithms for Three-Dimensional Bin Packing

2025-03-11

AI Summary

This paper studies three NP-hard axis-aligned non-overlapping packing problems for 3D rectangular boxes: 3D bin packing (3D-BP), 3D strip packing (3D-SP), and 3D minimum-volume bounding box (3D-MVBB). We propose a novel framework integrating harmonic segmentation, hierarchical packing, and volume scaling, augmented by an enhanced classification strategy and progressive analysis techniques. Our contributions include: (i) the first asymptotic polynomial-time approximation scheme (APTAS) for 3D-MVBB; (ii) improved absolute approximation ratios of 6 for 3D-BP, 6 for 3D-SP, and 3+ε for 3D-MVBB; and (iii) an improved asymptotic approximation ratio of ≈2.54 for 3D-BP. All results substantially surpass the best-known bounds since 1990, establishing new state-of-the-art absolute and asymptotic approximation ratios under both models.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Computer Vision: 3D Computer Vision
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

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 $T_{infty}^2 + varepsilonapprox 2.86$, 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.

Problem

• Develops improved approximation algorithms for 3D Bin Packing.
• Enhances solutions for 3D Strip Packing with minimal height.
• Optimizes Minimum Volume Bounding Box packing efficiency.

Innovation

• Improved approximation ratios for 3D packing problems
• Asymptotic approximation ratio of 2.54 for 3D-BP
• APTAS for Minimum Volume Bounding Box