We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to derive dual programs for problems which could be formulated as convex assignment problems. The duals have intuitive structures as the ones in linear programming. The constraints of the duals explicitly indicate the online decisions and naturally lead to competitive algorithms. Second, we use a dual-fitting approach, which also based on the weak duality, to study problems which are unlikely to admit convex relaxations. Through the analysis, we show an interesting feature in which primal-dual gives idea for designing algorithms while the analysis is done by dual-fitting. We illustrate the advantages and the flexibility of the approaches through problems in different setting: from single machine to unrelated machine environments, from typical competitive analysis to the one with resource augmentation, from convex relaxations to non-convex relaxations.
This article examines two different mechanisms for saving power in battery-operated embedded systems. The first strategy is that the system can be placed in a sleep state if it is idle. However, a fixed amount of energy is required to bring the system back into an active state in which it can resume work. The second way in which power savings can be achieved is by varying the speed at which jobs are run. We utilize a power consumption curve P(s) which indicates the power consumption level given a particular speed. We assume that P(s) is convex, nondecreasing, and nonnegative for s ≥ 0. The problem is to schedule arriving jobs in a way that minimizes total energy use and so that each job is completed after its release time and before its deadline. We assume that all jobs can be preempted and resumed at no cost. Although each problem has been considered separately, this is the first theoretical analysis of systems that can use both mechanisms. We give an offline algorithm that is within a factor of 2 of the optimal algorithm. We also give an online algorithm with a constant competitive ratio. We consider the problem of scheduling on a single processor a given set of n jobs. Each job j has a workload wj and a release time r j. The processor can vary its speed and hibernate to reduce energy consumption. In a schedule minimizing overall consumed energy, it might be that some jobs complete arbitrarily far from their release time. So in order to guarantee some quality of service, we would like to impose a deadline d j = rj + F for every job j, where F is a guarantee on the flow time. We provide an O(n3) algorithm for the more general case of agreeable deadlines, where jobs have release times and deadlines and can be ordered such that for every i < j, both ri rj and di dj. We study an energy conservation problem where a variable-speed processor is equipped with a sleep state. Executing jobs at high speeds and then setting the processor asleep is an approach that can lead to further energy savings compared to standard dynamic speed scaling. We consider classical deadline-based scheduling, i.e. each job is specified by a release time, a deadline and a processing volume. For general convex power functions, [12] devised an offline 2-approximation algorithm. Roughly speaking, the algorithm schedules jobs at a critical speed Scrit that yields the smallest energy consumption while jobs are processed. For power functions P(s) = sα + γ, where s is the processor speed, [11] gave an (αα + 2)-competitive online algorithm. We investigate the offline setting of speed scaling with a sleep state. First we prove NP-hardness of the optimization problem. Additionally, we develop lower bounds, for general convex power functions: No algorithm that constructs Scrit-schedules, which execute jobs at speeds of at least scrit, can achieve an approximation factor smaller than 2. Furthermore, no algorithm that minimizes the energy expended for processing jobs can attain an approximation ratio smaller than 2. We then present an algorithmic framework for designing good approximation algorithms. For general convex power functions, we derive an approximation factor of 4 3. For power functions P(s) = βsα + γ, we obtain an approximation of 137 117 < 1.171. We finally show that our framework yields the best approximation guarantees for the class of Scrit-schedules. For general convex power functions, we give another 2-approximation algorithm. For functions P(s) = βsα + γ, we present tight upper and lower bounds on the best possible approximation factor. The ratio is exactly eW−1(−e−1−1 e) (eW−1(−e−1−1 e) + 1) < 1.211, where W--1 is the lower branch of the Lambert W function.