We study the problem of allocating indivisible resources under the connectivity constraints of a graph $G$. This model, initially introduced by Bouveret et al. (published in IJCAI, 2017), effectively encompasses a diverse array of scenarios characterized by spatial or temporal limitations, including the division of land plots and the allocation of time plots. In this paper, we introduce a novel fairness concept that integrates local comparisons within the social network formed by a connected allocation of the item graph. Our particular focus is to achieve pairwise-maximin fair share (PMMS) among the "neighbors" within this network. For any underlying graph structure, we show that a connected allocation that maximizes Nash welfare guarantees a $(1/2)$-PMMS fairness. Moreover, for two agents, we establish that a $(3/4)$-PMMS allocation can be efficiently computed. Additionally, we demonstrate that for three agents and the items aligned on a path, a PMMS allocation is always attainable and can be computed in polynomial time. Lastly, when agents have identical additive utilities, we present a pseudo-polynomial-time algorithm for a $(3/4)$-PMMS allocation, irrespective of the underlying graph $G$. Furthermore, we provide a polynomial-time algorithm for obtaining a PMMS allocation when $G$ is a tree.

翻译：我们研究在连通约束图$G$下分配不可分割资源的问题。该模型最初由Bouveret等人（发表于IJCAI, 2017）提出，有效涵盖了具有空间或时间限制的多种场景，包括地块划分和时间片段分配。在本文中，我们引入了一个新的公平性概念，该概念在物品图连通分配所形成的社会网络中整合了局部比较。我们特别关注在此网络中实现“邻居”间的成对最大最小公平份额（PMMS）。对于任意底层图结构，我们证明最大化纳什福利的连通分配能保证$(1/2)$-PMMS公平性。此外，对于两智能体，我们证明可高效计算出$(3/4)$-PMMS分配。进一步，对于三智能体且物品排列在路径上的情况，我们证明PMMS分配总是可达的，且可在多项式时间内计算。最后，当智能体具有相同可加效用时，我们提出一个伪多项式时间算法以获得$(3/4)$-PMMS分配，该算法适用于任意底层图$G$。此外，当$G$为树时，我们提供多项式时间算法以获得PMMS分配。