Scheduling coupled tasks on parallel identical machines

In many applications in the context of patient appointment scheduling there are recurring tasks with fixed delays between them. These tasks are commonly referred to as coupled-task jobs. In the coupled-task settings, each job consists of two tasks whereby the second task must start processing after an exact time lag following the completion of the first task. In this paper, we introduce the problem of scheduling a set of coupled-task jobs on parallel identical machines with the objective function of minimizing the makespan. We study the computational complexity of the general problem, as well as its special cases. We prove that the majority of these problems are (strongly) NP-hard. Nonetheless, we provide the optimal scheduling policy for two settings consisting of identical jobs. An important result of our work includes showing that the existence of a (2-e)-approximation algorithm for the problem implies P=NP. The latter result improves a recently proposed bound for the open-shop counterpart as well.