In this paper, we consider multiprocessor scheduling with submodular penalties to extend multiprocessor scheduling with rejection to submodular function. An instance of the problem is given by n jobs and m machines with each job having a certain processing time on a machine. We aim to find a subset R of rejected jobs, and assign each of other jobs to one of the m machines. The objective is to minimize the sum of the makespan of the m machines and the rejection penalty R, where the rejection penalty is determined by a submodular function. For this problem, we design a non-combinatorial Lovász rounding algorithm that achieves a worst-case guarantee of \(\frac{3+\sqrt{5}}{2}\). Then, we consider a special case of this problem in which all the machines are identical, i.e. each job has the same processing time on any machine, and we design a combinatorial \((2-\frac{1}{m})\)-approximation algorithm based on the greedy method and list scheduling (LS) algorithm.

在本文中，我们考虑了带有亚模块惩罚的多处理器调度，以扩展拒绝子模块功能的多处理器调度。该问题的一个实例由n个作业和m台计算机给出，每个作业在计算机上具有一定的处理时间。我们的目标是找到拒绝的作业的子集R，并将其他每个作业分配给m台机器之一。目的是最小化m个机器的制造时间和拒绝惩罚R的总和，其中拒绝惩罚由子模函数确定。针对此问题，我们设计了一种非组合式Lovász舍入算法，该算法可实现最坏情况的保证\（\ frac {3+ \ sqrt {5}} {2} \）。然后，我们考虑这个问题的特例，其中所有机器都相同，即每个作业在任何机器上具有相同的处理时间，并且我们设计了组合\（（2- \ frac {1} {m}）\ ） -基于贪婪方法和列表调度（LS）算法的近似算法。