In this paper, we study the parameterized complexity of several variants of scheduling with precedence constraints between jobs. Namely, we consider the single machine setting with delay values on top of the precedence constraints. Such scheduling problems are related to several decades-old problems with open parameterized complexity status, notably Shuffle Product and Directed Bandwidth. We obtain XNLP-completeness results for both problems, and derive implications to scheduling with minimum (resp. maximum) delays parameterized by the width of the directed acyclic graph giving the precedence constraints, and/or by the maximum delay value in the input. Regarding Directed Bandwidth, we also settle the case of trees by showing XNLP-completeness parameterized by the target value. Beyond these results, we believe that Shuffle Product is an unusual and promising addition to the list of XNLP-complete problems.

翻译：本文研究了具有作业间优先约束的多种调度变体的参数复杂性。具体而言，我们考虑在优先约束之上附加延迟值的单机设置。此类调度问题与数个参数复杂性状态悬而未决的经典问题相关，尤其是混洗乘积和有向带宽问题。我们证明了这两个问题均属于XNLP完全类，并推导出其对以有向无环图宽度（定义优先约束）和/或输入最大延迟值为参数的极小（或极大）延迟调度问题的结论。针对有向带宽问题，我们通过证明以目标值为参数的XNLP完全性，进一步确定了树形结构的情况。除这些结果外，我们认为混洗乘积是XNLP完全问题列表中一个独特且富有前景的新成员。