We study the following variant of the classic {\em bin packing} problem. Given a set of items of various sizes, partitioned into groups, find a packing of the items in a minimum number of identical (unit-size) bins, such that no two items of the same group are assigned to the same bin. This problem, known as {\em bin packing with clique-graph conflicts}, has natural applications in storing file replicas, security in cloud computing and signal distribution. Our main result is an {\em asymptotic polynomial time approximation scheme (APTAS)} for the problem, improving upon the best known ratio of $2$. %In particular, for any instance $I$ and a fixed $\eps \in (0,1)$, the items are packed in at most $(1+\eps)OPT(I) +1$ bins, where $OPT(I)$ is the minimum number of bins required for packing the instance. As a key tool, we apply a novel {\em Shift \& Swap} technique which generalizes the classic linear shifting technique to scenarios allowing conflicts between items. The major challenge of packing {\em small} items using only a small number of extra bins is tackled through an intricate combination of enumeration and a greedy-based approach that utilizes the rounded solution of a {\em linear program}.

翻译：我们研究经典 {em bin 包装} 问题的以下变量。 鉴于一系列不同大小的项目, 被分成组, 在一个相同( unit- size) bin 的最小数量( unit- size) bin 中找到一个项目包装, 也就是说没有将同一个组的两个项目分配给同一个文件夹。 这个问题被称为 em bin bin 和 cloi- graphy- ground}, 具有存储文件复制件、 云计算和信号分布中的安全性等自然应用。 我们的主要结果就是对问题采用一种无拘无束的多元时间近似方案( APTAS ), 改进已知的$2美元的最佳比率。% 特别是, 无论如何, 美元和固定 $\ eeps\ 等于 ( 0, 1, 1) 。 这个问题在存储文件复制件、 云计算和信号分布中, $OPT (I) 是一个最起码的包装件数。 作为关键工具, 我们应用一种新型的 Shift & Swap} 技术, 将一个典型的直线性方案, 将一个典型的宏一小的宏的宏的宏的包装组合, 组合中, 将一个方案将一个小的宏的宏的宏的宏的宏的宏的线性的方法, 将它用于在使用一个简单的组合。