Since its development in the early 90's, parameterized complexity has been widely used to analyze the tractability of many NP-hard combinatorial optimization problems with respect to various types of problem parameters. While the generic nature of the framework allows the analysis of any combinatorial problem, the main focus along the years was on analyzing graph problems. In this paper we diverge from this trend by studying the parameterized complexity of Just-In-Time (JIT) flow-shop scheduling problems. Our analysis focuses on the case where the number of due dates is considerably smaller than the number of jobs, and can thus be considered as a parameter. We prove that the two-machine problem is W[1]-hard with respect to this parameter, even if all processing times on the second machine are of unit length, while the problem is in XP even for a parameterized number of machines. We then move on to study the tractability of the problem when combining the different number of due dates with either the number of different weights or the number of different processing times on the first machine. We prove that in both cases the problem is fixed-parameter tractable for the two machine case, and is W[1]-hard for three or more machines.

中文翻译：

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准时制流水车间调度问题的参数化易处理性

自 90 年代初发展以来，参数化复杂度已被广泛用于分析许多 NP-hard 组合优化问题相对于各种类型的问题参数的易处理性。虽然该框架的通用性质允许分析任何组合问题，但多年来的主要重点是分析图问题。在本文中，我们通过研究准时制 (JIT) 流水车间调度问题的参数化复杂性来偏离这一趋势。我们的分析侧重于截止日期数量远小于工作数量的情况，因此可以将其视为一个参数。我们证明双机问题对于这个参数是 W[1]-hard，即使第二台机器上的所有处理时间都是单位长度，而问题出在 XP 中，即使是参数化数量的机器。然后，我们继续研究将不同到期日数与不同权重数或第一台机器上不同处理时间数相结合时问题的易处理性。我们证明，在这两种情况下，问题对于两台机器来说都是固定参数可处理的，而对于三台或更多台机器来说则是 W[1]-hard。