A note on reverse scheduling with maximum lateness objective

Abstract

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 $$1||\sum \alpha _{j}T_{j}^{2}$$ .