We revisit the unrelated machine scheduling problem with the weighted completion time objective. It is known that independent rounding achieves a 1.5 approximation for the problem, and many prior algorithms improve upon this ratio by leveraging strong negative correlation schemes. On each machine $i$, these schemes introduce strong negative correlation between events that some pairs of jobs are assigned to $i$, while maintaining non-positive correlation for all pairs. Our algorithm deviates from this methodology by relaxing the pairwise non-positive correlation requirement. On each machine $i$, we identify many groups of jobs. For a job $j$ and a group $B$ not containing $j$, we only enforce non-positive correlation between $j$ and the group as a whole, allowing $j$ to be positively-correlated with individual jobs in $B$. This relaxation suffices to maintain the 1.5-approximation, while enabling us to obtain a much stronger negative correlation within groups using an iterative rounding procedure: at most one job from each group is scheduled on $i$. We prove that the algorithm achieves a $(1.36 + \epsilon)$-approximation, improving upon the previous best approximation ratio of $1.4$ due to Harris. While the improvement may not be substantial, the significance of our contribution lies in the relaxed non-positive correlation condition and the iterative rounding framework. Due to the simplicity of our algorithm, we are able to derive a closed form for the weighted completion time our algorithm achieves with a clean analysis. Unfortunately, we could not provide a good analytical analysis for the quantity; instead, we rely on a computer assisted proof.

翻译：本文重新研究了以加权完工时间为目标的无关机调度问题。已知独立舍入方法对该问题可实现1.5近似比，而许多现有算法通过引入强负相关方案进一步改进该比率。在每台机器$i$上，这些方案在部分作业对被分配到$i$的事件之间建立强负相关性，同时保证所有作业对之间保持非正相关性。本文算法通过放宽成对非正相关性要求而偏离此传统方法：在每台机器$i$上，我们识别出多个作业组。对于作业$j$及其不隶属的作业组$B$，我们仅要求$j$与整个作业组之间保持非正相关性，允许$j$与$B$内单个作业存在正相关性。这种松弛条件足以维持1.5近似比，同时使我们能通过迭代舍入程序在组内实现更强的负相关性：每台机器$i$上至多调度每个作业组中的一个作业。我们证明该算法能达到$(1.36 + \epsilon)$近似比，改进了Harris先前提出的最佳近似比1.4。虽然改进幅度有限，但本研究的核心贡献在于松弛的非正相关性条件与迭代舍入框架。得益于算法简洁性，我们通过清晰分析推导出算法所得加权完工时间的闭合表达式。遗憾的是，我们未能对该量值给出完善的理论分析，转而采用计算机辅助证明进行验证。