AI 中文总结
研究装箱问题变体qBP，因经典FFD算法证明假设不适用于此问题，故通过分析精心选择子实例上的FFDq(D_q)规避困难，得出在某些特殊情况下，FFDq(D_q)的近似比率为FFDq(D_q) ≤ 11/9 OPT(D_q) + 3q。
AI 中文摘要
我们考虑一种装箱问题的变体，对每个物品的副本数量及其在装箱中的放置有约束。输入\(D_q := DD\ldots\)被定义为多重集\(D\)的\(q\)个连续副本，具有固定的箱容量\(S\)。目标是将\(D_q\)中的所有物品装入最少数量的箱中，使得每个箱中每个物品最多包含一个副本，且箱中所有物品的总大小不超过箱容量\(S\)，我们称此问题为\(q\)BP。首次适应递减（\(\mathsf{FFD}\)）是一种经典的装箱算法。在文献中，\(\mathsf{FFD}\)的证明依赖于\(\mathsf{FFD}\)装箱中最后一个箱只包含一个物品的假设，此假设不适用于\(q\)BP问题。本文通过在精心选择的子实例\({D'}_q \subseteq D_q\)上分析\(\mathsf{FFDq(D_q)}\)（\(D\)的\(q\)个连续副本，每个副本按非递增顺序排序）来规避此困难，同时保持对原始输入\(D_q\)相同的上界。我们表明，对于某些特殊情况，\(\mathsf{FFDq(D_q)}\)的近似比率为\(\mathsf{FFDq(D_q)} \leq \frac{11}{9}\mathsf{OPT(D_q)} + 3q\)，其中\(\mathsf{FFDq}\)和\(\mathsf{OPT}\)分别表示\(\mathsf{FFD}\)泛化和最优算法使用的箱数。
英文摘要
We consider a variant of the bin packing problem with constraints on the number of copies of each item and their placement in the packing. The input $D_q := DD\ldots$ is defined as $q$ consecutive copies of the multiset $D$, with a fixed bin capacity $S$. Note that, for each item in $D$, there are $q$ copies in $D_q$. The goal is to pack all the items in $D_q$ into the minimum number of bins, such that each bin contains at most one copy of each item and the total size of all items in a bin does not exceed the bin capacity $S$. We call this problem $q$BP. First Fit Decreasing ($\mathsf{FFD}$) is a classical bin packing algorithm: it first orders the items in nonincreasing order, then packs the next item into the first bin where it fits. In the literature, $\mathsf{FFD}$ proofs rely on the assumption that the last bin in the $\mathsf{FFD}$ packing contains only a single item. This assumption does not naturally extend to the $q$BP problem. In this paper, we circumvent this difficulty by analyzing $\mathsf{FFDq(D_q)}$ on a carefully chosen subinstance ${D'}_q \subseteq D_q$ ($q$ consecutive copies of $D$, each copy sorted in non-increasing order) while preserving the same upper bound for the original input $D_q$. We show that the approximation ratio of $\mathsf{FFDq(D_q)}$ for some special cases is \begin{align*} \mathsf{FFDq(D_q)} \leq \frac{11}{9}\mathsf{OPT(D_q)} + 3q \end{align*} where $\mathsf{FFDq}$ and $\mathsf{OPT}$ denote the number of bins used by the $\mathsf{FFD}$ generalization and by an optimal algorithm, respectively.
Comments14 pages; work in progress
