  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.
