抢占式调度中一个开放问题的结构性质
Structural Properties of an Open Problem in Preemptive Scheduling
摘要：
在最近的一些论文中，已经研究了最优抢占式调度的结构特性，主要集中在两个结构参数上：所需的最小抢占次数，以及“移位”的严格下限，即，由抢占、作业开始或完成所创建的时间所限定的间隔的大小。到目前为止，只有粗糙的界限，这些参数已被导出的具体问题。本文针对抢占式调度理论中的一个著名的公开问题：由带有释放日期的$n$单位执行时间工件的入树组成的调度问题，以极小化两个处理机上的总完成时间为目标，给出了这些结构参数的界.这是目前，诱人的“阈值”问题的调度理论：我们的文献调查显示，任何重大的推广导致NP难问题，但任何重大的简化导致易处理的问题。对于上述问题，我们证明了最优性所需的抢占次数不需要超过$2n-1$ ;在某些情况下，抢占次数必须是$Ω(\log n)$阶;最小移位不需要小于$2^{-2n+1}$。这些界限是通过组合分析的最佳时间表，而不是通过分析的多面体角落的线性规划配方，一种方法可以在早期的论文中找到。边界立即遵循一个基本的结构属性称为“正常”，其中最小的变化的工作是指数下降的功能。特别地，被抢占作业的开始与其抢占之间的第一间隔是1/2的倍数，第二个这样的间隔是1/4的倍数，并且通常，第$i$个抢占发生在$2^{-i}$的倍数处。我们希望新的结构特性在最终解决一个令人烦恼的、仍然悬而未决的复杂性问题方面发挥重要作用。
英文摘要：
Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on `shifts', i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. So far only rough bounds for these parameters have been derived for specific problems. This paper sharpens the bounds on these structural parameters for a well-known open problem in the theory of preemptive scheduling: Instances consist of in-trees of $n$ unit-execution-time jobs with release dates, and the objective is to minimize the total completion time on two processors. This is among the current, tantalizing `threshold' problems of scheduling theory: Our literature survey reveals that any significant generalization leads to an NP-hard problem, but that any significant simplification leads to tractable problem. For the above problem, we show that the number of preemptions necessary for optimality need not exceed $2n-1$; that the number must be of order $Ω(\log n)$ for some instances; and that the minimum shift need not be less than $2^{-2n+1}$. These bounds are obtained by combinatorial analysis of optimal schedules rather than by the analysis of polytope corners for linear-program formulations, an approach to be found in earlier papers. The bounds immediately follow from a fundamental structural property called `normality', by which minimal shifts of a job are exponentially decreasing functions. In particular, the first interval between a preempted job's start and its preemption is a multiple of 1/2, the second such interval is a multiple of 1/4, and in general, the $i$-th preemption occurs at a multiple of $2^{-i}$. We expect the new structural properties to play a prominent role in finally settling a vexing, still-open question of complexity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
