We study the problem of minimizing total weighted completion time on unrelated parallel machines with machine-independent job weights. The best previous approximation guarantee for this problem is arbitrarily close to 1.36, due to Li [SODA 2025], who developed a configuration-LP and iterative-rounding framework based on randomly shifted geometric size classes and a computer-assisted final analysis. We improve the approximation guarantee to arbitrarily close to 1.3168. The scheduling algorithm retains Li's configuration-LP and iterative-rounding framework, with a different fixed geometric class ratio. The improvement comes from a new analysis of the random geometric shift. For each machine and Smith prefix, we normalize the physical prefix before averaging over the shift, so that its normalized size distribution and configuration moments remain fixed across all shifts. A scale-dependent configuration bound and a cross-shift averaging argument then reduce the approximation analysis to a one-dimensional certificate. The final certificate is computer assisted and reproducible. Its rational data are verified exactly where possible, while the remaining continuous inequalities are certified using directed interval arithmetic. This yields the first improvement over Li's guarantee for the machine-independent-weight model.

翻译：暂无翻译