Proportionally Fair Makespan Approximation

TL;DR

The paper studies fair mechanism design for makespan minimization in scheduling on unrelated machines, introducing a mean-efficiency–based characterization that ties proportional allocations to efficiency. It presents the Anti-Diagonal Mechanism, a polynomial-time procedure that converts any input allocation into a proportionable one with a $3/2$-approximation to the optimal makespan, and proves this bound tight for general instances; for normalized costs, it achieves the optimal makespan. The work also extends to the goods allocation dual, showing no proportional mechanism can guarantee any $\beta$-approximation in general, while achieving optimal egalitarian welfare in normalized settings. Finally, it analyzes relaxations of envy-freeness, showing that approximate envy-freeness can circumvent known lower bounds, including the development of cyclic-envy-free mechanisms, and outlines open directions for constant-factor approximations under envy relaxations and for normalized instances.

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 $Ω(\log m/ \log \log m)$-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.

Figures (1)

• Figure 1: Illustration of the Anti-Diagonal Mechanism for the case where $m = n$, with the initial allocation $B_i = \{i\}$ and the parameter $k$ set to $1$. Rows represent machines and columns represent jobs. Empty dots indicate the initial allocation $B$. The subfigures show the allocation matrix after the swap and merge operations, where full dots represent the resulting allocation.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 2.1: General and normalized instances
• Definition 2.2: Mechanisms
• Definition 2.3: Makespan approximation
• Definition 2.4: Proportional mechanism
• Definition 2.5: Proportionable allocation