无关机器加权完成时间的交叉偏移分析：(1.3168+ε)-近似算法
Cross-Shift Analysis for Unrelated-Machine Weighted Completion Time A (1.3168+epsilon)-Approximation
AI总结：
本研究针对与机器无关权重的无关并行机总加权完成时间问题，改进近似保证至接近1.3168，通过调整几何类比率优化配置-LP与迭代舍入框架，经交叉偏移分析及计算机辅助证书验证实现突破。
AI中文摘要：
我们研究在与机器无关的作业权重的无关并行机上最小化总加权完成时间的问题。该问题此前最优的近似保证可任意接近1.36，由Li在[SODA 2025]中提出，其开发了基于随机偏移几何尺寸类和计算机辅助最终分析的配置线性规划（configuration-LP）与迭代舍入框架。我们将近似保证改进为可任意接近1.3168。调度算法保留了Li的配置线性规划与迭代舍入框架，仅采用不同的固定几何类比率。改进源于对随机几何偏移的新分析：对每台机器和Smith前缀，我们在偏移平均前归一化物理前缀，使其归一化大小分布和配置矩在所有偏移间保持固定；随后，基于尺度的配置界与交叉偏移平均论证将近似分析简化为一维证书。最终证书为计算机辅助且可复现，其有理数据在可能处被精确验证，其余连续不等式用定向区间算术认证，这是与机器无关权重模型中对Li保证的首次改进。
英文摘要：
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.
