SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1649-1669, January 2020.
In this paper we consider one of the most basic scheduling problems where jobs have their respective arrival times and deadlines. The goal is to schedule as many jobs as possible nonpreemptively by their respective deadlines on $m$ identical parallel machines. For the last decade, the best approximation ratio known for the single-machine case ($m = 1$) has been $1-1/e - \epsilon \approx 0.632$ due to Chuzhoy, Ostrovsky, and Rabani [FOCS 2001] and [MOR 2006]. We break this barrier and give an improved 0.644-approximation. For the multiple-machine case, we give an algorithm whose approximation guarantee becomes arbitrarily close to 1 as the number of machines increases. This improves upon the previous best $1 - 1/ (1 + 1/m)^m$ approximation due to Bar-Noy et al. [STOC 1999] and [SICOMP 2009], which converges to 1-1/e as $m$ goes to infinity. Our result for the multiple-machine case extends to the weighted throughput objective where jobs have different weights, and the goal is to schedule jobs with the maximum total weight. Our results show that the 1 - 1/e approximation factor widely observed in various coverage problems is not tight for the nonpreemptive maximum throughput scheduling problem.

中文翻译：

---

打破1/1 / e壁垒，实现非抢先吞吐量最大化

SIAM离散数学杂志，第34卷，第3期，第1649-1669页，2020年1月。
在本文中，我们考虑了最基本的调度问题之一，其中作业具有各自的到达时间和截止时间。目标是在$ m $相同的并行计算机上按其各自的截止日期，尽早地安排尽可能多的作业。在过去的十年中，由于Chuzhoy，Ostrovsky和Rabani [FOCS 2001]和[e]，单机情况下最好的近似比率（$ m = 1 $）为1-1 / e-\ epsilon \大约0.632 $。 [MOR 2006]。我们打破了这一障碍，并给出了改进的0.644近似值。对于多机情况，我们给出了一种算法，其近似保证随着机数的增加而变得接近于1。由于Bar-Noy等人的观点，这在先前的最佳$ 1-1 /（1 + 1 / m）^ m $近似值的基础上有所改善。[STOC 1999]和[SICOMP 2009]，随着$ m $达到无穷大，收敛到1-1 / e。对于多机案例，我们的结果扩展到加权吞吐量目标，其中作业具有不同的权重，目标是安排最大总重量的作业。我们的结果表明，对于非抢占式最大吞吐量调度问题，在各种覆盖问题中广泛观察到的1-1 / e近似因子并不严格。