Note on on-line scheduling with precedence constraints on identical machines

We consider the on-line scheduling problem of jobs with precedence constraints on m parallel identical machines. Each job has a time processing requirement, and may depend on other jobs (has to be processed after them). A job arrives only after its predecessors have been completed. The cost of an algorithm is the time at which the last job is completed. We show lower bounds on the competitive ratio of on-line algorithms for this problem in several versions. We prove a lower bound of 2-1/m on the competitive ratio of any deterministic algorithm (with or without preemption) and a lower bound of 2-2/(m+1) on the competitive ratio of any randomized algorithm (with or without preemption). The lower bounds for the cases that preemption is allowed require arbitrarily long sequences. If we use only sequences of length O(m²), we can show a lower bound of 2-2/(m+1) on the competitive ratio of deterministic algorithms with preemption, and a lower bound of 2-O(1/m) on the competitive ratio of any randomized algorithm with preemption. All the lower bounds hold even for sequences of unit jobs only. The best algorithm that is known for this problem is the well-known List Scheduling algorithm of Graham. The algorithm is deterministic and does not use preemption. The competitive ratio of this algorithm is 2-1/m. Our randomized lower bounds are very close to this bound (a difference of O(1/m)) and our deterministic lower bounds match it.