Improved Approximation Algorithms for Three-Dimensional Knapsack
Klaus Jansen, Debajyoti Kar, Arindam Khan, K. V. N. Sreenivas, Malte Tutas
TL;DR
The paper presents a breakthrough in approximating the 3D Knapsack problem by introducing container packing, a structured approach that partitions the knapsack into a constant number of containers and reduces inside-container packing to tractable subproblems via GAP. A central structural lemma shows OPT can be bounded by a multiple of the profit of a guessable container packing, enabling a polynomial-time PTAS for box packing with a ratio of 139/29 (plus epsilon). The authors extend the framework to rotations, achieving a 30/7 (+ε) approximation, and derive improved results for cardinality and uniform profit-density variants; they also introduce new container types (notably L-Containers) to handle complex 3D packings. Together, these results push 3DK closer to practical, provable guarantees, with the container approach offering a versatile template for other multidimensional packing problems. The work also establishes hardness barriers for the container-family approach, highlighting scope for further refinement.
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, Thöle, 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 $\varepsilon>0$. Chlebík and Chlebíková (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).