Keeping the Harmony Between Neighbors: Local Fairness in Graph Fair Division

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.