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 α-approximation algorithms in the resource augmentation setting, i.e., finding a feasible partition on machines running at speed α ≥ 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, λ, is bounded by a constant. However, their algorithm has a super-exponential dependence on λ and hence does not extend to larger values of λ. Our main contribution is to design an approximation scheme with a substantially improved running-time dependence on λ. In particular, our algorithm depends exponentially on logλ and hence has quasi-polynomial running time even if λ 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.