Improved Approximation Algorithms for Three-Dimensional Knapsack

2025-03-25

AI Summary

This paper studies the 3D knapsack problem (3DK): packing a maximum-profit subset of axis-aligned boxes into a unit cube without overlap. We introduce the “container packing” paradigm—partitioning items into a constant number of structured containers and designing geometry-specific packing strategies for each container type. Our approach integrates generalized assignment problem (GAP) modeling, dynamic programming, and polynomial-time enumeration. This yields the first improvement over the previous best approximation ratio of 5, achieving a ratio of (139/29 + ε) ≈ 4.794 for 3DK; with item rotation allowed, the ratio further improves to (30/7 + ε) ≈ 4.286. The framework extends naturally to variants with cardinality constraints and uniform-density items. Our results significantly advance the state-of-the-art in approximation algorithms for geometric 3D knapsack problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the three-dimensional Knapsack (3DK) problem, in which we are given a set of axis-aligned cuboids with associated profits and an axis-aligned cube knapsack. The objective is to find a non-overlapping axis-aligned packing (by translation) of the maximum profit subset of cuboids into the cube. The previous best approximation algorithm is due to Diedrich, Harren, Jansen, Th""{o}le, and Thomas (2008), who gave a $(7+varepsilon)$-approximation algorithm for 3DK and a $(5+varepsilon)$-approximation algorithm for the variant when the items can be rotated by 90 degrees around any axis, for any constant $varepsilon>0$. Chleb'{i}k and Chleb'{i}kov'{a} (2009) showed that the problem does not admit an asymptotic polynomial-time approximation scheme. We provide an improved polynomial-time $(139/29+varepsilon) approx 4.794$-approximation algorithm for 3DK and $(30/7+varepsilon) approx 4.286$-approximation when rotations by 90 degrees are allowed. We also provide improved approximation algorithms for several variants such as the cardinality case (when all items have the same profit) and uniform profit-density case (when the profit of an item is equal to its volume). Our key technical contribution is container packing -- a structured packing in 3D such that all items are assigned into a constant number of containers, and each container is packed using a specific strategy based on its type. We first show the existence of highly profitable container packings. Thereafter, we show that one can find near-optimal container packing efficiently using a variant of the Generalized Assignment Problem (GAP).

Problem

• Improving approximation algorithms for 3D knapsack packing
• Enhancing packing efficiency with container-based strategies
• Addressing variants like cardinality and uniform profit-density cases

Innovation

• Improved polynomial-time approximation algorithm for 3DK
• Container packing strategy for structured 3D packing
• Efficient near-optimal packing via Generalized Assignment Problem