A common approximation framework for the early work, the late work, and resource leveling problems with unit time jobs

We study the approximability of two related machine scheduling problems. In the late work minimization problem, there are identical parallel machines and the jobs have a common due date. The objective is to minimize the late work, defined as the sum of the portion of the jobs done after the due date. A related problem is the maximization of the early work, defined as the sum of the portion of the jobs done before the due date. We describe a polynomial time approximation scheme for the early work maximization problem, and we extended it to the late work minimization problem after shifting the objective function by a positive value that depends on the problem data. We also prove an inapproximability result for the latter problem if the objective function is shifted by a constant which does not depend on the input. These results remain valid even if the number of the jobs assigned to the same machine is bounded. This leads to an extension of our approximation scheme to some variants of the resource leveling problem, for which no approximation algorithms were known.