Time-dependent scheduling with linear deterioration involves determining when to execute jobs whose processing times degrade as their beginning is delayed. Each job i is associated with a release time r_i and a processing time function p_i(s_i)=alpha_i + beta_i*s_i, where alpha_i, beta_i>0$ are constants and s_i is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. Here, we take a step forward by developing new bounds and approximation results for the interesting special case of the problems with uniform deterioration, i.e.\ beta_i=beta, for each i. The key contribution is a O(1+1/beta)-approximation algorithm for the makespan problem and a O(1+1/beta^2)-approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with beta=O(1/n) and beta=Omega(n), where n is the number of jobs. Our analysis is based on a new approach for comparing computed and optimal schedules via bounding pseudomatchings.

翻译：时间依赖线性劣化调度涉及确定何时执行那些处理时间随开始时间延迟而增加的任务。每个任务i与一个释放时间r_i和一个处理时间函数p_i(s_i)=α_i+β_i*s_i相关联，其中α_i、β_i>0为常数，s_i为任务的开始时间。在此设定下，单机最小化最大完工时间和总完工时间问题的可近似性仍待解决。本文通过针对均匀劣化（即对所有i有β_i=β）这一有趣特例问题开发新边界和近似结果而取得进展。关键贡献包括：针对最大完工时间问题提出O(1+1/β)近似算法，针对总完工时间问题提出O(1+1/β²)近似算法。此外，我们针对β=O(1/n)和β=Ω(n)的实例（其中n为任务数量）提出了贪心常数因子近似算法。我们的分析基于一种通过界定伪匹配来比较计算调度与最优调度的新方法。