In this paper, we work on the scheduling problem with active time model. We have a set of preemptive jobs with integral release times, deadlines and required processing lengths, while the preemption of jobs is only allowed at integral time points. We have a single machine that can process at most g distinct job units at any given time unit when the machine is switched on. The objective is to find a schedule that completes all jobs within their timing constraints and minimizes the time when the machine is on, i.e., the active time. This problem has been studied by Chang et al. where they proposed an LP rounding approach which gives a 2-approximation solution. In this paper, we also give a 2-approximation algorithm based on LP rounding approach with a different rounding technique and analysis. Finally, we give a new linear programming formulation for which we conjecture that the integrality gap is 5/3, which might bring new hope for beating the barrier of 2 for the approximation ratio.

在本文中，我们使用活动时间模型研究调度问题。我们有一组抢占式作业，具有完整的发布时间，截止日期和所需的处理时间，而抢占式作业仅在完整的时间点才被允许。我们有一台机器最多可以处理g开机时，在任何给定时间单位都可以使用不同的作业单位。目的是找到一个时间表，该时间表可在其时间限制内完成所有作业，并最大程度地缩短机器开机的时间（即活动时间）。Chang等人已经研究了这个问题。他们提出了LP舍入方法，该方法给出了2个近似解。在本文中，我们还给出了一种基于LP舍入方法的2近似算法，并采用了不同的舍入技术和分析方法。最后，我们给出了一个新的线性规划公式，我们推测其完整性差距为5/3，这可能为克服近似值的2的障碍带来新的希望。