This paper considers parallel machine scheduling with incompatibilities between jobs. The jobs form a graph and no two jobs connected by an edge are allowed to be assigned to the same machine. In particular, we study the case where the graph is a collection of disjoint cliques. Scheduling with incompatibilities between jobs represents a well-established line of research in scheduling theory and the case of disjoint cliques has received increasing attention in recent years. While the research up to this point has been focused on the makespan objective, we broaden the scope and study the classical total completion time criterion. In the setting without incompatibilities, this objective is well known to admit polynomial time algorithms even for unrelated machines via matching techniques. We show that the introduction of incompatibility cliques results in a richer, more interesting picture. Scheduling on identical machines remains solvable in polynomial time, while scheduling on unrelated machines becomes APX-hard. Furthermore, we study the problem under the paradigm of fixed-parameter tractable algorithms (FPT). In particular, we consider a problem variant with assignment restrictions for the cliques rather than the jobs. We prove that it is NP-hard and can be solved in FPT time with respect to the number of cliques. Moreover, we show that the problem on unrelated machines can be solved in FPT time for reasonable parameters, e.g., the parameter pair: number of machines and maximum processing time. The latter result is a natural extension of known results for the case without incompatibilities and can even be extended to the case of total weighted completion time. All of the FPT results make use of n-fold Integer Programs that recently have received great attention by proving their usefulness for scheduling problems.
翻译:本文审视了与工作不相容的平行机器日程安排。 任务将形成一个图表, 不允许向同一机器分配任何两个带有边缘连接的工作。 特别是, 我们研究图表中包含不相容的线索。 任务之间不相兼容的线索显示的是列表理论的既定研究路线, 与不相兼容的线索案例近年来越来越受到越来越多的关注。 虽然直到这个点的研究一直集中在 makespan 目标上, 我们扩大了任务范围, 并研究了典型的完全完成时间标准。 在不兼容的设置中, 这个目标被广为人知的是, 接受混合时间算法, 甚至通过匹配技术, 甚至不相干不相干的机器。 我们显示, 引入不兼容的线索是更丰富、 更有趣的画面。 在不相关的机器上, APX 的进度也变得非常困难。 此外, 我们研究的是, 固定和可调频调的算算法模式下的问题。 特别是, 在不相容的设置不相容的情况下, 使用不相干断的数学时间算法的计算结果, 也证明, 在不易变法的 Ralalalal rolate rolate ad rodudududududuc roduc ro roduc rodu rodudududududududududuc 。