Complexity results for scheduling tasks with discrete starting times |
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Authors: | Kazuo Nakajima S. Louis Hakimi |
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Affiliation: | Computer Science Section, Department of Electrical Engineering, Texas Tech University, Box 4439, Lubbock, Texas 79409 USA;Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60201 USA |
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Abstract: | Suppose that n independent tasks are to be scheduled without preemption on a set of identical parallel processors. Each task Ti requires a given execution time τi and it may be started for execution on any processor at any of its prescribed starting times si1, si2, …, siki, with ki ≤ 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 τi ε {τ, τ′} and ki ≤ 3 for 1 ≤ i ≤ n. The same problem is, however, shown to be solvable in O(n log n) time, provided siki − si1 < τi 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 τi ε {τ, τ′}, ki = 2 and si2 − si1 = δ < τi for 1 ≤ i ≤ n. Finally a special case with ki = 2 and si2 − si1 = 1, 1 ≤ i ≤ n, of the above multiprocessor scheduling problem is shown to be solvable in polynomial time. |
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