Proportionally Fair Makespan Approximation

TL;DR

This paper introduces a proportional mechanism with payments that achieves a tight 3/2-approximation for fair job scheduling on unrelated machines, contrasting with the limitations of envy-free mechanisms.

Contribution

It presents the first proportional mechanism with payments that guarantees a tight approximation ratio for makespan minimization, along with a full characterization of proportional allocation functions.

Findings

Proportional mechanisms can achieve a 3/2-approximation, which is tight.

Envy-free mechanisms cannot surpass an  log m / log log m approximation.

For normalized costs, the optimal makespan can be achieved by proportional mechanisms.

Abstract

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.