A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem $1||L_{\max }$ are studied in 2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: $\ell _{1},\ell _{2},\ell _{\infty },\ell _{H}^{\Sigma } $ , and $\ell _{H}^{\max }$ . It appears that the $O(n^{2})$ -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms $\ell _{1},\ell _{\infty },\ell _{H}^{\Sigma }$ , and $ \ell _{H}^{\max }$ , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm $\ell _{2}$ is treated separately. As a by-product, we resolve an open question on the complexity of problem $1||\sum \alpha _{j}T_{j}^{2}$ .