We study a scheduling problem where the jobs we have to perform are composed of one or more tasks. If two jobs sharing a non-empty subset of tasks are scheduled on the same machine, then these shared tasks have to be performed only once. This kind of problem is known in the literature under the names of VM-PACKING or PAGINATION. Our objective is to schedule a set of these objects on two parallel identical machines, with the aim of minimizing the makespan. This problem is NP-complete as an extension of the PARTITION problem. In this paper we present three exact algorithms with worst-case time-complexity guarantees, by exploring different branching techniques. Our first algorithm focuses on the relation between jobs sharing one or more symbols in common, whereas the two other algorithms branches on the shared symbols.

我们研究一个调度问题，其中我们必须执行的工作由一个或多个任务组成。如果在同一台计算机上调度了共享非空任务子集的两个作业，则这些共享任务仅需执行一次。这种问题在文献中以VM-PACKING或PAGINATION的名称为人所知。我们的目标是在两台平行的相同机器上安排一组这些对象，以最大程度地减少工期。作为PARTITION的扩展，此问题是NP完全的问题。在本文中，我们通过探索不同的分支技术，提出了三种具有最坏情况时间复杂度保证的精确算法。我们的第一个算法着眼于共享一个或多个符号的作业之间的关系，而其他两个算法则基于共享的符号。