Fixed-server-time complexity classification

Determine whether the two-machine scheduling problem with one common loading–unloading server and fixed equal loading and unloading time s admits a pseudo-polynomial algorithm or is strongly NP-hard for some fixed value of s.

Background

The paper proves that the decision version of P2,S1|s_j=t_j=s|C_max is NP-complete for every fixed integer s≥1 and strongly NP-complete when s is part of the input. However, the reduction used for fixed s is from PARTITION and does not establish strong NP-hardness. The authors therefore leave unresolved whether the fixed-s problem has a pseudo-polynomial algorithm or is strongly NP-hard for at least one fixed server-operation duration.

References

The theorem concerns the case in which s is part of the input. It does not settle whether the fixed-s problem is strongly NP-hard for some fixed value of s or admits a pseudo-polynomial algorithm.

— Minimizing the Makespan Approximately on Two Identical Parallel Machines with a Loading--Unloading Server  (2608.18837 - Hasani et al., 19 Aug 2026) in Section 3, immediately after Theorem 2