We study the computational complexity of fair division of indivisible items in an enriched model: there is an underlying graph on the set of items. And we have to allocate the items (i.e., the vertices of the graph) to a set of agents in such a way that (a) the allocation is fair (for appropriate notions of fairness) and (b) each agent receives a bundle of items (i.e., a subset of vertices) that induces a subgraph with a specific "nice structure." This model has previously been studied in the literature with the nice structure being a connected subgraph. In this paper, we propose an alternative for connectivity in fair division. We introduce what we call compact graphs, and look for fair allocations in which each agent receives a compact bundle of items. Through compactness, we try to capture the idea that every agent must receive a set of "closely related" items. We prove a host of hardness and tractability results for restricted input settings with respect to fairness concepts such as proportionality, envy-freeness and maximin share guarantee.

翻译：我们研究了在增强模型中不可分割物品公平分配的计算复杂性：物品集合上存在一个底层图。我们需要将物品（即图的顶点）分配给一组代理人，使得（a）分配是公平的（基于适当的公平性概念），且（b）每个代理人获得一个诱导出具有特定“良好结构”子图的物品束（即顶点子集）。该模型此前在文献中已有研究，其中良好结构为连通子图。本文提出了公平分配中连通性的替代方案。我们引入了所谓的紧凑图，并寻求每个代理人获得紧凑物品束的公平分配。通过紧凑性，我们试图体现每个代理人必须获得一组“紧密相关”物品的思想。针对比例公平性、无嫉妒性和最大最小份额保证等公平性概念，我们证明了一系列受限输入设置下的难解性与可解性结果。