We consider a recently introduced fair repetitive scheduling problem involving a set of clients, each asking for their associated job to be daily scheduled on a single machine across a finite planning horizon. The goal is to determine a job processing permutation for each day, aiming to minimize the maximum total completion time experienced by any client. This problem is known to be NP-hard for quite restrictive settings, with previous work offering exact solution methods for highly-structured special cases. In this paper, we focus on the design of approximation algorithms with provable performance guarantees. Our main contributions can be briefly summarized as follows: (i) When job processing times are day-dependent, we devise a polynomial-time LP-based $2$-approximation, as well as a polynomial-time approximation scheme for a constant number of days. (ii) With day-invariant processing times, we obtain a surprisingly simple $(\frac{1+\sqrt{2}}{2}+ε)$-approximation in polynomial time. This setting is also shown to admit a quasi-polynomial-time approximation scheme for an arbitrary number of days. The key technical component driving our approximation schemes is a novel batching technique, where jobs are conceptually grouped into batches, subsequently leading either to a low-dimensional dynamic program or to a compact configuration LP. Concurrently, while developing our constant-factor approximations, we propose a host of lower-bounding mechanisms that may be of broader interest.

翻译：我们研究一个近期提出的公平重复调度问题，该问题涉及一组客户，每位客户要求其关联的作业在有限规划周期内每日在一台机器上进行调度。目标是确定每日的作业处理顺序，以最小化任意客户所经历的最大总完成时间。已知该问题在相当严格的设定下是NP难的，先前的研究仅为高度结构化的特殊情形提供了精确求解方法。本文聚焦于设计具有可证明性能保证的近似算法。我们的主要贡献可简要概括如下：(i) 当作业处理时间与日期相关时，我们设计了一种基于线性规划的多项式时间$2$-近似算法，以及针对固定天数的多项式时间近似方案。(ii) 在处理时间与日期无关的情况下，我们在多项式时间内获得了一个令人惊讶的简单$(\frac{1+\sqrt{2}}{2}+ε)$-近似算法。该设定还被证明对于任意天数均允许拟多项式时间近似方案。驱动我们近似方案的关键技术组件是一种新颖的批处理技术，其中作业在概念上被分组为批次，随后导向低维动态规划或紧凑配置线性规划。同时，在构建常数因子近似算法的过程中，我们提出了一系列可能具有更广泛研究价值的下界构造机制。