We study vector bin packing and vector bin covering problems, multidimensional generalizations of the classical bin packing and bin covering problems, respectively. In 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 $B$, we have $\left\|\sum_{v\in B}v\right\|_\infty\leq 1$. 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. Hence, 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 $\frac{600}{599}$, for $d=2$. 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 $B$, we have $\sum_{v\in B}v\geq \mathbf 1$. 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:我们研究的矢量 bin 包装和 矢量 bin bin bin bin 分别包含问题、 经典 bin 包装和 bin 的多维通用 。 在 矢量 Bin 包装中, 我们得到了一套美元( 0. 1美元) 的 美元维度矢量矢量矢量, 目的是将集分成最小数的垃圾箱。 事实上, 我们显示, 对于每件 bin $B$, 我们得到了$\ sumv\ v\ in B} vright\ nbty\ leq 1美元 。 Woodginger [Woe97] 声称, 问题没有 APTAS 的维度大于或等于 2. 我们注意到原始证据中存在轻微监督。 因此, 我们用一些来自 [ BKKS06, CC09] 的更多想法给出了经过修改的证明。 事实上, 我们很难得到比 $\600, 600599 美元 更好的 。 在矢量 的矢量 中, 我们得到了比 $ 01\ binal= bin 最高的直值 。