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:对冲突( BPC) 的垃圾包装进行综合研究。 输入是一个固定的美元项目、 大小( 美元) 、 大小( 0. 1美元) 和 一个冲突图形( 美元) = ( I, E) 。 我们的目标是找到一个以最小数量独立的数据集( 每个总大小以$为单位) 的美元分隔。 作为臭名昭著的硬图形颜色问题的概观, BPC 大部分被研究在多色化的冲突图表中。 一个令人感兴趣的问题就是, 这些图表中的 BPC 是否接受与经典的 bin 包装相同的已知近似保证 : I\ 0. 1美元 美元 和 一个冲突图解 。 我们通过显示( 与 bin 包装相对的) BPC 最小化的多元化时间接近方案( APTAS ) 已经出现在表面上简单易懂的图形类中, 如双色图解 和 平面图解中。 我们用一个新式的 1.3- $- approupal clas for the explain ligroupal exgraphol2, a ligal_ a ligill ligill ligal ligill ligroupstal ligal listolum) 。