We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an $\Omega(\log m/ \log \log m)$-approximation to the optimal makespan, where $m$ is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a $3/2$-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.

翻译：我们研究了经典作业调度问题在无关机器上的公平机制，其目标是最小化完工时间。该问题等价于在公平分配杂务时最小化平等主义社会成本。公平分配文献中两种主流的公平性概念是无嫉妒性和比例性。先前的研究已经证明，即使允许向机器支付费用，任何无嫉妒性机制也无法提供优于 $\Omega(\log m/ \log \log m)$ 近似比的方案来逼近最优完工时间，其中 $m$ 是机器数量。与此不可能性形成鲜明对比的是，我们的主要结果表明，存在一种比例性机制（带支付）能够实现 $3/2$ 近似比逼近最优完工时间，并且该比率是紧的。为了证明这一结果，我们完整刻画了可以通过支付实现比例性的分配函数类别。此外，我们证明对于成本已归一化的实例，存在一种比例性机制可以达到最优完工时间。最后，我们提出了关于其他公平性概念（包括无嫉妒性的松弛形式）的未来重要研究方向。值得注意的是，我们证明了导致无嫉妒性不可能结果的技术并不能推广到其松弛形式。