Pseudo-polynomial solvability for fixed-period maximal fairness

Determine whether the fixed-period maximal fairness problem in repetitive single-machine scheduling is solvable in pseudo-polynomial time when the quality-of-service criterion is total completion time, waiting time, lateness, or tardiness, including the cases known to be weakly NP-hard for every fixed constant q≥3 under completion time and every fixed constant q≥2 under the other three criteria.

Background

The paper studies the maximal fairness problem, denoted by 1|rep|max_j∑i f{ij}(C_{ij}), in which q daily single-machine schedules must be chosen to minimize the largest aggregate quality-of-service value received by any client. The relevant quality-of-service measures are total completion time, waiting time, lateness, and tardiness.

For an arbitrary number of days, the problem is strongly NP-hard for all four criteria. With q fixed, the paper summarizes known weak NP-hardness thresholds: q≥3 for completion time and q≥2 for waiting time, lateness, and tardiness. The unresolved issue is whether these fixed-q weakly NP-hard cases nevertheless admit pseudo-polynomial algorithms.