Scheduling problems involving a set of dependent tasks with release dates and deadlines on a limited number of resources have been intensively studied. However, few parameterized complexity results exist for these problems. This paper studies the existence of a feasible schedule for a typed task system with precedence constraints and time intervals \((r_i,d_i)\) for each job i. The problem is denoted by \(P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert \star \). Several parameters are considered: the pathwidth pw(I) of the interval graph I associated with the time intervals \((r_i, d_i)\), the maximum processing time of a task \(p_{\max }\) and the maximum slack of a task \(s\ell _{\max }\). This paper establishes that the problem is para-\(\textsf{NP}\)-complete with respect to any of these parameters. It then provides a fixed-parameter algorithm for the problem parameterized by both parameters pw(I) and \(\min (p_{\max },s\ell _{\max })\). It is based on a dynamic programming approach that builds a levelled graph which longest paths represent all the feasible solutions. Fixed-parameter algorithms for the problems \(P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert C_{\max }\) and \(P\vert \mathcal{M}_j(type),prec,r_i\vert L_{\max }\) are then derived using a binary search.

涉及有限资源上的一组具有发布日期和截止日期的相关任务的调度问题已经得到深入研究。然而，这些问题的参数化复杂性结果很少。本文研究了对于每个作业 i具有优先约束和时间间隔\((r_i,d_i)\)的类型化任务系统的可行调度的存在性。该问题由\(P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert \star \)表示。考虑几个参数：与时间间隔\((r_i, d_i)\)关联的区间图I的路径宽度pw ( I ) 、任务的最大处理时间\(p_{\max }\)以及任务的最大松弛度\(s\ell _{\max }\)。本文确定该问题对于任何这些参数都是半完整的。然后，它为由参数pw ( I ) 和\(\min (p_{\max },s\ell _{\max })\)参数化的问题提供固定参数算法。它基于动态编程方法，构建一个分层图，其中最长路径代表所有可行的解决方案。问题\(P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert C_{\max }\)和\(P\vert \mathcal{M}_j(type) 问题的固定参数算法),prec,r_i\vert L_{\max }\)然后使用二分搜索导出。