We study the problem of maximizing the throughput of jobs wherein each job consists of multiple tasks. Consider a system offering a capacity of one unit. We are given a set of jobs, each consisting of a sequence of $r$ tasks. Each task is associated with a demand and an interval where it should be scheduled. Each job has a profit associated with it. The objective is to select a subset of jobs having the maximum profit such that at any point of time the cumulative demand of the selected jobs does not exceed the system capacity. Prior work has presented $O\left(r\right)$-approximation algorithm for two special cases: (i) unit demands – all tasks have a demand of one unit; (ii) uniform demands – for any job, all its constituent tasks have the same demand. We study the general setting wherein the tasks of a job can have non-uniform demands. We present an $O\left(r\right)$-approximation for the case where all the tasks have demands at most $1∕2$. The prior algorithms are based on the fractional local ratio technique, which we do not know how to generalize for the setting of non-uniform demands. Instead, we devise an algorithm based on the randomized rounding strategy. Our approach also provides alternative simpler algorithms for the unit and uniform demand settings. We next extend the algorithms to the more generic scenario wherein each task is associated with a processing time and a window consisting of release time and deadline, and design pseudo-polynomial time $O\left(r\right)$-approximation procedures.

我们研究最大化作业吞吐量的问题，其中每个作业包含多个任务。考虑一个提供一个单位容量的系统。我们得到了一组工作，每个工作由一系列$\mathrm{[R}$任务。每个任务都与需求和应计划的时间间隔相关联。每个工作都有与之相关的利润。目的是选择利润最大的作业子集，以使所选作业的累积需求在任何时间点都不会超过系统容量。先前的工作已经提出$Ø（\mathrm{[R}）$-两种特殊情况的近似算法：（i）单位需求–所有任务的需求都为一个单位；（ii）统一的需求-对于任何工作，其所有组成任务都具有相同的需求。我们研究了一项工作任务可能具有不一致要求的一般设置。我们提出一个$Ø（\mathrm{[R}）$-对于所有任务最多具有需求的情况的近似值 $\mathrm{1个}∕2$。现有的算法基于分数局部比率技术，我们不知道如何针对不均匀需求的设置进行概括。相反，我们设计了一种基于随机舍入策略的算法。我们的方法还为单位和统一需求设置提供了其他更简单的算法。接下来，我们将算法扩展到更通用的场景，其中每个任务都与处理时间和由发布时间和截止日期组成的窗口以及设计伪多项式时间相关联$Ø（\mathrm{[R}）$-近似程序。