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.

中文翻译：

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不兼容派系调度的总完成时间最小化

本文考虑了作业之间不兼容的并行机器调度。作业形成一个图形，并且不允许将边连接的两个作业分配给同一台机器。特别地，我们研究了图形是不相交集团的集合的情况。作业之间不兼容的调度代表了调度理论中一条完善的研究路线，近年来，不相交集团的情况受到越来越多的关注。虽然到目前为止的研究都集中在完工时间目标上，但我们扩大了范围并研究了经典的总完工时间标准。在没有不兼容的情况下，这个目标是众所周知的，即使对于不相关的机器，也可以通过匹配技术承认多项式时间算法。我们表明，不兼容派系的引入会产生更丰富、更有趣的画面。在相同的机器上调度仍然可以在多项式时间内解决，而在不相关的机器上调度变得 APX-hard。此外，我们在固定参数易处理算法（FPT）的范式下研究了这个问题。特别是，我们考虑了一个问题变体，其中有派系而不是工作的分配限制。我们证明它是 NP 难的，并且可以在 FPT 时间内解决关于派系数量的问题。此外，我们表明，对于合理的参数，例如参数对：机器数量和最大处理时间，可以在 FPT 时间内解决无关机器上的问题。后一个结果是已知结果的自然扩展，没有不兼容的情况，甚至可以扩展到总加权完成时间的情况。所有 FPT 结果都使用了 n-fold Integer Programs，这些程序最近通过证明它们对调度问题的有用性而受到了极大的关注。