Energy-Efficient Algorithms for Non-preemptive Speed-Scaling

Abstract

We improve complexity bounds for energy-efficient non-preemptive scheduling problems for both the single processor and multi-processor cases. As energy conservation has become a major concern, traditional scheduling problems have been revisited in the past few years to take into account the energy consumption [1]. We consider the speed scaling setting introduced by Yao et al. [20] where a set of jobs, each with a release date, deadline and work volume, are to be scheduled on a set of identical processors. The processors may change speed as a function of time and the energy they consume is the $\alpha$th power of their speed integrated over time. The objective is then to find a feasible non-preemptive schedule which minimizes the total energy used.

We show that for an arbitrarily number of processors and jobs with equal work volumes there is a $2(1+\varepsilon )(5(1+\varepsilon ))^{\alpha -1}\tilde{B}_{\alpha }=O_{\alpha }(1)$ approximation algorithm, where $\tilde{B}_{\alpha }$ is the generalized Bell number. This is the first constant factor algorithm for the multi-processor case, and this also extends to arbitrary processor-dependent work volumes, up to losing a factor of $(\frac{(1+r)r}{2})^{\alpha }$ in the approximation, where $r$ is the maximum ratio between two work volumes. For the single processor case, we introduce a new linear programming formulation of speed scaling, using a new constraint capturing non-preemption, and prove that its integrality gap is at most $12^{\alpha -1}$. With our new constraint we improve on the previously known unbounded integrality gap of at least $\varOmega (n^{\alpha -1})$. Finally, we deal with the inapproximabilty of speed scaling and we prove that the multi-processor case is APX-hard, even in the special case where all release dates and deadlines are equal and $r$ is 4.

I. Milis—Partially supported by the project THALES-ALGONOW co-financed by the European Union (European Social Fund - ESF) and Greek national funds, through the Operational Program “Education and Lifelong Learning”.