Scheduling in the two-machine flowshop with due date constraints

The problem of minimizing either the maximum tardiness or the number of tardy jobs in a two-machine flowshop with due date considerations is known to be strongly NP-hard. A recent paper presented polynomial-time reductions from Partition to some restricted cases and concluded their ordinary NP-hardness. However, no pseudo-polynomial-time algorithm was developed. In this paper, we conduct polynomial-time reductions from 3-Partition to these special cases. The results confirm the strong NP-hardness of the problems under study. We also identify some other special cases as polynomially solvable by presenting their solution methods.