Complexity Results for Scheduling Tasks with Discrete Starting Times

Abstract Suppose that n independent tasks are to be scheduled without preemption on a set of identical parallel processors. Each task T i requires a given execution time τ i and it may be started for execution on any processor at any of its prescribed starting times s i 1 , s i 2 , …, s ik i , with k i ≤ k for some fixed integer k . We first prove that the problem of finding a feasible schedule on a single processor is NP-complete in the strong sense even when τ i ϵ { τ , τ ′} and k i ≤ 3 for 1 ≤ i ≤ n . The same problem is, however, shown to be solvable in O( n log n ) time, provided s ik i − s i 1 τ i for 1 ≤ i ≤ n . We then show that the problem of finding a feasible schedule on an arbitrary number of processors is strongly NP-complete even when τ i ϵ { τ , τ ′}, k i = 2 and s i 2 − s i 1 = δ τ i for 1 ≤ i ≤ n . Finally a special case with k i = 2 and s i 2 − s i 1 = 1, 1 ≤ i ≤ n , of the above multiprocessor scheduling problem is shown to be solvable in polynomial time.