Coordination Mechanisms for Weighted Sum of Completion Times in Machine Scheduling

We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. In short, we present the first constant-factor-approximate coordination mechanisms for this model and show that our bounds imply a new combinatorial constant-factor approximation algorithm for the underlying optimization problem. More specifically: We present a generalization of the ShortestFirst policy for weighted jobs, called SmithRule policy; we prove that it achieves an approximation ratio of 4 and show that any set of strongly local ordering policies can result in equilibria with approximation ratio at least 4 even for unweighted jobs. The main result of our paper is ProportionalSharing, a preemptive strongly local policy that generalizes the EqualSharing policy for weighted jobs and beats this lower bound of 4; we show that this policy achieves an approximation ratio of 2.619 for the weighted sum of completion times and an approximation ratio of 2.5 for the (unweighted) sum of completion times. Again, we observe that these bounds are tight. Furthermore, we show that the ProportionalSharing policy induces potential games (in which best-response dynamics converge to pure Nash equilibria). All of our upper bounds are for the robust price of anarchy, defined by Roughgarden [39], so they naturally extend to mixed Nash equilibria, correlated equilibria, and regret minimization dynamics. Finally, we prove that the games induced by ProportionalSharing are beta?-nice, which yields the first combinatorial constant-factor approximation algorithm minimizing weighted completion time for unrelated machine scheduling.