Revisit the scheduling problem with assignable or generalized due dates to minimize total weighted late work

We revisit the single-machine scheduling for minimising the total weighted late work with assignable due dates (ADD-scheduling) and generalised due dates (GDD-scheduling). In particular, we consider the following three problems: (i) the GDD-scheduling problem for minimising the total weighted late work, (ii) the ADD-scheduling problem for minimising the total weighted late work, and (iii) the ADD-scheduling problem for minimising the total late work. In the literature, the above three problems are proved to be NP-hard, but their exact complexity (unary NP-hardness or pseudo-polynomial-time solvability) are unknown. In this paper, we address these open problems by showing that the first two problems are unary NP-hard and the third problem admits pseudo-polynomial-time algorithms. For the third problem, we also present a 2-approximation solution and a fully polynomial-time approximation scheme. Computational experiments show that our algorithms and solutions are efficient. When the jobs have identical processing times, we further present more efficient polynomial-time algorithms.