Nearly Tight Bounds for Proportional Group Fair Divisions and One-Sided Discrepancy

2026-09-03

AI Summary

本文研究了不可分割物品在k个群体间的公平分配问题，通过改进方法将最坏情况下的偏差优化至约sqrt(n/k)，并为此开发了新的不等式工具。

Abstract

This paper studies the problem of fair division of indivisible goods among $k$ groups of $n_1,\ldots, n_k$ agents. We look at the worst downward deviation $\textit{PROP}(n_1,\ldots, n_k)$ of an agent in a group from its $1/k$-share. We improve the bounds of (Manurangsi and Meka, 2026) and show that $\textit{PROP}(n_1,\ldots, n_k) = \tildeΘ(\sqrt{n/k})$, where $n = n_1 + \ldots + n_k$ is the total number of agents. For the proof of the upper bound, we develop novel discrepancy-type tools and, in particular, a way to efficiently work with one-sided discrepancy constraints.

Problem

• fair division
• indivisible goods
• discrepancy

Innovation

• Proportional Group Fair Divisions
• One-Sided Discrepancy
• Discrepancy-Type Tools