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Personnel scheduling problems have received considerable academic attention due to their relevance in various real-world applications. These problems involve preparing feasible schedules for an organization's employees and often account for factors such as qualifications of workers and holiday requests, resulting in complex constraints. While certain versions of the personnel rostering problem are widely acknowledged as NP-hard, there is limited theoretical analysis specific to many of its variants. Many studies simply assert the NP-hardness of the general problem without investigating whether the specific cases they address inherit this computational complexity. In this paper, we examine a variant of the personnel scheduling problems, which involves scheduling a homogeneous workforce subject to constraints concerning both the total number and the number of consecutive work days and days off. This problem was claimed to be NP-complete by [Brunner+2013]. In this paper, we prove its NP-completeness and investigate how the combination of constraints contributes to this complexity. Furthermore, we analyze various special cases that arise from the omission of certain parameters, classifying them as either NP-complete or polynomial-time solvable. For the latter, we provide easy-to-implement and efficient algorithms to not only determine feasibility, but also compute a corresponding schedule. |
