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 \(\frac{998}{997}\).