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We study the approximability of two related machine scheduling problems. Inthe late work minimization problem, there are identical parallel machines andthe 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. Arelated problem is the maximization of the early work, defined as the sum ofthe portion of the jobs done before the due date. We describe a polynomial timeapproximation scheme for the early work maximization problem, and we extendedit to the late work minimization problem after shifting the objective functionby a positive value that depends on the problem data. We also prove aninapproximability result for the latter problem if the objective function isshifted by a constant which does not depend on the input. These results remainvalid 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 theresource leveling problem, for which no approximation algorithms were known.

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