Preemptive parallel-machine scheduling with a common server to minimize makespan

We consider parallel-machine scheduling with a common server and job preemption to minimize the makespan. While the non-preemptive version of the problem is strongly NP-hard, the complexity status of the preemptive version has remained open. We show that the preemptive version is NP-hard even if there is a fixed number of machines. We give a pseudo-polynomial time algorithm to solve the case with two machines. We show that the case with an arbitrary number of machines is unary NP-hard, analyze the performance ratios of some natural heuristic algorithms, and present several solvable special cases.