Improved approximation schemes for early work scheduling on identical parallel machines with common due date

We study the early work scheduling problem on identical parallel machines in order to maximize the total early work, i.e., the parts of non-preemptive jobs executed before a common due date. By preprocessing and constructing an auxiliary instance which has several good properties, we propose an efficient polynomial time approximation scheme with running time $O(n)$, which improves the result in [Gy{o}rgyi, P., Kis, T. (2020). A common approximation framework for early work, late work, and resource leveling problems. {it European Journal of Operational Research}, 286(1), 129-137], and a fully polynomial time approximation scheme with running time $O(n)$ when the number of machines is a fixed number, which improves the result in [Chen, X., Liang, Y., Sterna, M., Wang, W., B{l}a.{z}ewicz, J. (2020b). Fully polynomial time approximation scheme to maximize early work on parallel machines with common due date. {it European Journal of Operational Research}, 284(1), 67-74], where $n$ is the number of jobs, and the hidden constant depends on the desired accuracy.