Approximating Unrelated Machine Weighted Completion Time Using Iterative Rounding and Computer Assisted Proofs

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. Nevertheless, the checking algorithm for the analysis is easy to implement, essentially involving evaluation of maximum values of single-variable quadratic functions over given intervals. Therefore, unlike previous results which use intricate analysis to optimize the final approximation ratio, we delegate this task to computer programs.