Optimal-time implementation of LPT

Establish whether the Longest Processing Time First (LPT) heuristic for uniform machine scheduling admits an implementation with worst-case running time O((n+m)(log m + log n)) in the offline setting where all jobs are known in advance.

Background

The paper presents the first near-linear time implementation of the LPT heuristic with running time O((n+m)(log^2 m + log n)) by reducing assignment decisions to dynamic maintenance of the lower envelope of lines. While this improves upon the naive O(mn) approach, the authors point out a potential path to further optimization, motivated by the offline nature of the problem where all jobs are known upfront.

The explicit open problem asks whether the extra log m factor in the update time (from log^2 m to log m) can be eliminated to achieve an overall O((n+m)(log m + log n)) running time, thereby attaining what they consider an optimal, near-linear complexity for LPT implementation.