We give a comprehensive study of bin packing with conflicts (BPC). The input is a set $I$ of items, sizes $s:I \rightarrow [0,1]$, and a conflict graph $G = (I,E)$. The goal is to find a partition of $I$ into a minimum number of independent sets, each of total size at most $1$. Being a generalization of the notoriously hard graph coloring problem, BPC has been studied mostly on polynomially colorable conflict graphs. An intriguing open question is whether BPC on such graphs admits the same best known approximation guarantees as classic bin packing. We answer this question negatively, by showing that (in contrast to bin packing) there is no asymptotic polynomial-time approximation scheme (APTAS) for BPC already on seemingly easy graph classes, such as bipartite and split graphs. We complement this result with improved approximation guarantees for BPC on several prominent graph classes. Most notably, we derive an asymptotic $1.391$-approximation for bipartite graphs, a $2.445$-approximation for perfect graphs, and a $\left(1+\frac{2}{e}\right)$-approximation for split graphs. To this end, we introduce a generic framework relying on a novel interpretation of BPC allowing us to solve the problem via maximization techniques. Our framework may find use in tackling BPC on other graph classes arising in applications.

翻译：摘要：本文对带冲突装箱问题(BPC)进行了全面研究。输入为一个物品集合$I$、物品尺寸函数$s:I \rightarrow [0,1]$以及一个冲突图$G = (I,E)$。目标是找到$I$的一个划分，使得每个独立集的总尺寸至多为$1$，且独立集数量最小化。作为著名的图着色问题的难解变体，BPC的研究主要集中于多项式可着色的冲突图。一个引人注目的未解问题是：此类图上BPC问题是否能够获得与经典装箱问题相同的最佳已知近似保证。本文否定了该问题，证明即使在看似简单的图类（如二部图和分裂图）上，BPC也不存在渐近多项式时间近似方案(APTAS)（这有别于装箱问题）。针对若干典型图类，我们给出了改进的近似保证。最重要的是：针对二部图我们得到渐近$1.391$-近似，针对完美图得到$2.445$-近似，针对分裂图得到$\left(1+\frac{2}{e}\right)$-近似。为此，我们引入了一个通用框架，该框架基于对BPC问题的新颖解释，使我们能够通过最大化技术求解该问题。本框架可望应用于解决应用中其他图类上的BPC问题。