The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of $d$-dimensional rectangles, and the goal is to pack them into unit $d$-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when $d=2$. For general $d$, the best-known approximation algorithm has an approximation guarantee exponential in $d$, while the best hardness of approximation is still a small constant inapproximability from the case when $d=2$. In this paper, we show that the problem cannot be approximated within $d^{1-\epsilon}$ factor unless NP=ZPP. Recently, $d$-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within $\Omega(\log d)$ when $d$ is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when $d$ is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.

翻译：几何装箱问题（GBP）是装箱问题的一种推广，其输入为$d$维矩形的集合，目标是将它们高效地装入单位$d$维立方体中。即使当$d=2$时，该问题已不存在多项式时间近似方案（PTAS），属于NP难题。对于一般维数$d$，已知最佳近似算法的近似比随$d$指数增长，而近似难度下限仍仅为$d=2$情形下的小常数不可近似性。本文证明，除非NP=ZPP，否则该问题无法在$d^{1-\epsilon}$因子内近似。最近，与GBP密切相关的$d$维向量装箱问题，通过引入集合族的装箱维数概念，被证明在$d$为固定常数时难以在$\Omega(\log d)$因子内近似。本文提出其几何对应概念——几何装箱维数。尽管我们未能获得固定维数下几何装箱问题的类似不可近似性结果，但证明了几何装箱维数的若干关键性质，这些性质凸显了几何装箱与向量装箱之间的本质差异。