Approximating real-time scheduling on identical machines

We study the problem of assigning n tasks to m identical parallel machines in the real-time scheduling setting, where each task recurrently releases jobs that must be completed by their deadlines. The goal is to find a partition of the task set over the machines such that each job that is released by a task can meet its deadline. Since this problem is co-NP-hard, the focus is on finding a-approximation algorithms in the resource augmentation setting, i.e., finding a feasible partition on machines running at speed a¿=¿1, if some feasible partition exists on unit-speed machines. Recently, Chen and Chakraborty gave a polynomial-time approximation scheme if the ratio of the largest to the smallest relative deadline of the tasks, ¿, is bounded by a constant. However, their algorithm has a super-exponential dependence on ¿ and hence does not extend to larger values of ¿. Our main contribution is to design an approximation scheme with a substantially improved running-time dependence on ¿. In particular, our algorithm depends exponentially on log¿ and hence has quasi-polynomial running time even if ¿ is polynomially bounded. This improvement is based on exploiting various structural properties of approximate demand bound functions in different ways, which might be of independent interest.