Keep your distance: Land division with separation

Abstract

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 proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the 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.