We consider offline scheduling algorithms that incorporate speed scaling to address the bicriteria problem of minimizing energy consumption and a scheduling metric. For makespan, we give linear-time algorithms to compute all non-dominated solutions for the general uniprocessor problem and for the multiprocessor problem when every job requires the same amount of work. We also show that the multiprocessor problem becomes NP-hard when jobs can require different amounts of work. For total flow, we show that the optimal flow corresponding to a particular energy budget cannot be exactly computed on a machine supporting arithmetic and the extraction of roots. This hardness result holds even when scheduling equal-work jobs on a uniprocessor. We do, however, extend previous work by to give an arbitrarily-good approximation for scheduling equal-work jobs on a multiprocessor.
Minimizing the makespan of tasks with precedence constraints has also been studied in the context of project management. Speed scaling is possible when additional resources can be used to shorten some of the tasks. Pinedo @cite_0 gives heuristics for some variations of this problem.
@cite_0
2085836112 2097541335 2140916841 1440160528
We consider the problem of speed scaling to conserve energy in a multiprocessor setting where there are precedence constraints between tasks, and where the performance measure is the makespan. That is, we consider an energy bounded version of the classic problem Pm|prec|Cmax . We extend the standard 3-field notation and denote this problem as Sm|prec, energy|Cmax . We show that, without loss of generality, one need only consider constant power schedules. We then show how to reduce this problem to the problem Qm|prec|Cmax to obtain a poly-log(m)-approximation algorithm. We consider the problem of speed scaling to conserve energy in a multiprocessor setting where there are precedence constraints between tasks, and where the performance measure is the makespan. That is, we consider an energy bounded version of the classic problem Pm | prec | Cmax. We show that, without loss of generality, one need only consider constant power schedules. We then show how to reduce this problem to the problem Qm | prec | Cmax to obtain a poly-log(m)-approximation algorithm. We present polylogarithmic approximations for the R|prec|Cmax and R|prec|∑jwjCj problems, when the precedence constraints are “treelike” – i.e., when the undirected graph underlying the precedences is a forest. We also obtain improved bounds for the weighted completion time and flow time for the case of chains with restricted assignment – this generalizes the job shop problem to these objective functions. We use the same lower bound of “congestion+dilation”, as in other job shop scheduling approaches. The first step in our algorithm for the R|prec|Cmax problem with treelike precedences involves using the algorithm of Lenstra, Shmoys and Tardos to obtain a processor assignment with the congestion + dilation value within a constant factor of the optimal. We then show how to generalize the random delays technique of Leighton, Maggs and Rao to the case of trees. For the weighted completion time, we show a certain type of reduction to the makespan problem, which dovetails well with the lower bound we employ for the makespan problem. For the special case of chains, we show a dependent rounding technique which leads to improved bounds on the weighted completion time and new bicriteria bounds for the flow time. Resource-constrained project scheduling with the objective of minimizing project duration (RCPSP) is one of the most studied scheduling problems. In this paper we consider the RCPSP with general temporal constraints and calendar constraints. Calendar constraints make some resources unavailable on certain days in the scheduling period and force activity execution to be delayed while resources are unavailable. They arise in practice from, e.g., unavailabilities of staff during public holidays and weekends. The resulting problems are challenging optimization problems. We develop not only four different constraint programming (CP) models to tackle the problem, but also a specialized propagator for the cumulative resource constraints taking the calendar constraints into account. This propagator includes the ability to explain its inferences so it can be used in a lazy clause generation solver. We compare these models, and different search strategies on a challenging set of benchmarks using a lazy clause generation solver. We close 83 of the open problems of the benchmark set, and show that CP solutions are highly competitive with existing Mip models of the problem.