We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of $d$-dimensional vectors from $[0,1]^d$ and the aim is to partition the set into the minimum number of bins such that for each bin $B$, each component of the sum of the vectors in $B$ is at most 1. Woeginger [Woe97] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [BCKS06,CC09]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than $\frac{600}{599}$. An instance of Vector Bin Packing is called $\delta$-skewed if every item has at most one dimension greater than $\delta$. As a natural extension of our general $d$-Dimensional Vector Bin Packing result we show that for $\varepsilon\in (0,\frac{1}{2500})$ it is NP-hard to obtain a $(1+\varepsilon)$-approximation for $\delta$-Skewed Vector Bin Packing if $\delta>20\sqrt \varepsilon$. In the Vector Bin Covering problem given a set of $d$-dimensional vectors from $[0,1]^d$, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin $B$, each component of the sum of the vectors in $B$ is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than $\frac{998}{997}$.

翻译：我们研究向量装箱问题和向量装覆盖问题，它们分别是装箱问题和装覆盖问题的多维推广。在向量装箱问题中，给定一组来自$[0,1]^d$的$d$维向量，目标是将其划分为最少数量的箱子，使得对于每个箱子$B$，$B$中向量之和的每个分量至多为1。Woeginger [Woe97] 曾断言该问题在维度大于等于2时不存在APTAS。我们注意到原始证明中存在轻微疏漏。本文中，我们利用[BCKS06, CC09]中的一些额外思想，给出一个修订后的证明。实际上，我们证明获得优于$\frac{600}{599}$的渐近近似比是NP难的。若向量装箱问题的每个实例中每个物品至多有一个维度大于$\delta$，则称该实例为$\delta$-偏斜的。作为我们一般$d$维向量装箱问题结论的自然推广，我们证明对于$\varepsilon\in (0,\frac{1}{2500})$，若$\delta>20\sqrt \varepsilon$，则获得$\delta$-偏斜向量装箱问题的$(1+\varepsilon)$-近似是NP难的。在向量装覆盖问题中，给定一组来自$[0,1]^d$的$d$维向量，目标是得到具有最大基数的不相交子集族（称为箱子），使得对于每个箱子$B$，$B$中向量之和的每个分量至少为1。利用与向量装箱问题结论类似的思想，我们证明对于维度大于等于2的向量装覆盖问题，不存在APTAS。实际上，我们证明获得优于$\frac{998}{997}$的渐近近似比是NP难的。