Exact unrestricted LPT approximation ratio

Determine the exact worst-case performance ratio of the longest-processing-time rule for unrestricted instances of the two-machine scheduling problem with one common loading–unloading server and fixed equal loading and unloading time s≥2, and determine whether allowing short jobs increases the asymptotic worst-case ratio beyond the long-job limit of 11/8.

Background

For instances in which every processing time satisfies p_j≥s, the paper establishes the exact LPT ratio (11s−3)/(8s−2). For unrestricted instances, it obtains the lower and upper bounds (11s+2)/(8s+1) and (13s−2)/(8s), respectively. These bounds converge to 11/8 and 13/8 as s tends to infinity, so the exact unrestricted ratio and the effect of short jobs on the asymptotic ratio remain unresolved.

References

Determining whether short jobs genuinely increase the asymptotic worst-case ratio, and finding the exact value of \rho_{\mathrm{LPT}}(s) for fixed s\ge2, remain open.

— Minimizing the Makespan Approximately on Two Identical Parallel Machines with a Loading--Unloading Server  (2608.18837 - Hasani et al., 19 Aug 2026) in Section 7, Conclusion