Envy-Free Makespan Approximation

TL;DR

This paper develops envy-free scheduling mechanisms for unrelated machines, achieving near-optimal makespan approximations for indivisible tasks and optimal solutions for divisible tasks, advancing fair resource allocation methods.

Contribution

Introduces an envy-free polynomial-time mechanism with $O(\log m)$ approximation for indivisible tasks and proves existence of envy-free optimal makespan for divisible tasks.

Findings

Achieves $O(\log m)$ approximation for indivisible tasks.

Establishes lower bound of $\Omega(\log m / \log\log m)$ for envy-free approximation.

Proves existence of envy-free mechanism with optimal makespan for divisible tasks.

Abstract

We study envy-free mechanisms for scheduling tasks on unrelated machines (agents) that approximately minimize the makespan. For indivisible tasks, we put forward an envy-free poly-time mechanism that approximates the minimal makespan to within a factor of $O(\log m)$, where $m$ is the number of machines. We also show a lower bound of $\Omega(\log m / \log\log m)$. This improves the recent result of Hartline {\sl et al.} \cite{Ahuva:2008} who give an upper bound of $(m+1)/2$, and a lower bound of $2-1/m$. For divisible tasks, we show that there always exists an envy-free poly-time mechanism with optimal makespan.