This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of \emph{proportionality} is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the \emph{ordinal maximin share approximation}, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

翻译：本文是持续努力将公平分配理论更贴近实践的一部分，通过处理现实应用中的需求。我们聚焦于土地遗产划分中产生的两个需求：(1) 每个代理人应获得一块可用几何形状的地块，(2) 不同代理人的地块必须物理隔离。在这些需求下，经典的公平性概念"比例性"是不切实际的，因为可能无法实现其任何乘法近似。相比之下，Budish于2011年引入的"序数最大最小份额近似"提供了有意义的公平性保证。我们证明了当可用形状为正方形、粗矩形或任意轴对齐矩形时，可达到的最大最小份额保证的上界和下界，并探讨了在此设定下寻找公平划分的算法复杂性和查询复杂性。我们的工作利用了计算几何中的工具和概念，如矩形的独立集和断头台式划分。