Multiplicative guarantees for unshifted total weighted tardiness

Determine whether the single-machine total weighted tardiness problem with release dates and identical processing times admits a meaningful multiplicative approximation guarantee for the unshifted objective on restricted instance classes, including restrictions on release dates, job weights, weight-release orderings, or the number of release phases.

Background

The paper establishes strong NP-hardness for single-machine total weighted tardiness with release dates and identical processing times. It also analyzes a phase-grid assignment algorithm, but the proven multiplicative guarantee applies only to the shifted objective F+p∑jwjF+p\sum_j w_j, while the guarantee for ordinary total weighted tardiness FF is additive. The authors therefore leave open whether useful multiplicative guarantees can be obtained for the unshifted objective under additional structural restrictions on the instances.

References

Determining whether the unshifted problem admits a meaningful multiplicative guarantee on restricted instance classes, and whether other polynomial algorithms improve on the phase-grid assignment bound for the shifted objective, remain questions for future work.

— Strong NP-Hardness and Approximation Algorithm for Weighted Tardiness with Release Dates and Identical Processing Times  (2609.28751 - Chen et al., 23 Sep 2026) in Section Conclusion