Approximation Algorithms for Scheduling With/Without Deadline Constraints Where Rejection Costs are Proportional to Processing Times

Abstract

We study two offline job scheduling problems where tasks can be processed on a limited number of energy-efficient edge machines or offloaded to an unlimited supply of energy-inefficient cloud machines (called rejected). The objective is to minimize total energy consumption. First, we consider scheduling without deadlines, formulating it as a scheduling problem with rejection, where rejection costs are proportional to processing times. We propose a novel $\frac{5}{4}(1+\epsilon )$-approximation algorithm, $\mathcal {BEKP}$, by associating it to a Multiple Subset Sum problem, improving upon the existing $ (\frac{3}{2} - \frac{1}{2m})$-approximation for arbitrary rejection costs. Next, we address scheduling with deadlines, aiming to minimize the weighted number of rejected jobs. We position this problem within the literature and introduce a new $(1-\frac{(m-1)^{m}}{m^{m}})$-approximation algorithm, $\mathcal {MDP}$, inspired by an interval selection algorithm with a $(1-\frac{m^{m}}{(m+1)^{m}})$-approximation for arbitrary rejection costs. Experimental results demonstrate that $\mathcal {BEKP}$ and $\mathcal {MDP}$ obtain better results (lower costs or higher profits) than other state-of-the-art algorithms while maintaining a competitive or better time complexity.