Power-saving scheduling for weakly dynamic voltage scaling devices

We study the problem of non-preemptive scheduling to minimize energy consumption for devices that allow dynamic voltage scaling. Specifically, consider a device that can process jobs in a non-preemptive manner. The input consists of (i) the set R of available speeds of the device, (ii) a set J of jobs, and (iii) a precedence constraint Pi among J. Each job j in J, defined by its arrival time a(j), deadline d(j), and amount of computation c(j), is supposed to be processed by the device at a speed in R. Under the assumption that a higher speed means higher energy consumption, the power-saving scheduling problem is to compute a feasible schedule with speed assignment for the jobs in J such that the required energy consumption is minimized. This paper focuses on the setting of weakly dynamic voltage scaling, i.e., speed change is not allowed in the middle of processing a job. To demonstrate that this restriction on many portable power-aware devices introduces hardness to the power-saving scheduling problem, we prove that the problem is NP-hard even if a(j) = a(j') and d(j) = d(j') hold for all j,j' is an element of J and \R\ = 2. If \R\ < infinity, we also give fully polynomial-time approximation schemes for two cases of the general NP-hard problem: (a) all jobs share a common arrival time, and (b) Pi = 0 and for any j, j' is an element of J, a(j) <= a(j') implies d(j) <= d(j'). To the best of our knowledge, there is no previously known approximation algorithm for any special case of the NP-hard problem.