A $1.25(1+\epsilon )$-Approximation Algorithm for Scheduling with Rejection Costs Proportional to Processing Times

Abstract

We address an offline job scheduling problem where jobs can either be processed on a limited supply of energy-efficient machines, or offloaded to energy-inefficient machines (with an unlimited supply), and the goal is to minimize the total energy consumed in processing all tasks. This scheduling problem can be formulated as a problem of scheduling with rejection, where rejecting a job corresponds to process it on an energy-inefficient machine and has a cost directly proportional to the processing time of the job. To solve this scheduling problem, we introduce a novel $\frac{5}{4}(1+\epsilon )$ approximation algorithm $\mathcal {BEKP}$ by associating it to a Multiple Subset Sum problem. Our algorithm is an improvement over the existing literature, which provides a ($\frac{3}{2} - \frac{1}{2m}$) approximation for scheduling with arbitrary rejection costs. We evaluate and discuss the effectiveness of our approach through a series of experiments, comparing it to existing algorithms.