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

TL;DR

The paper studies fair division of indivisible items arranged on a graph under connectivity constraints, introducing a local fairness notion called pairwise maximin share ($PMMS$) that compares neighboring allocations in the social graph. It shows that a connected allocation maximizing Nash welfare guarantees a $1/2$-PMMS for any graph, and it provides efficient algorithms achieving $3/4$-PMMS for two agents, PMMS on a path with three agents, and a pseudo-polynomial-time solution for identical utilities on general graphs (with polynomial-time PMMS on trees). These results are complemented by a suite of structural techniques (bipolar orderings, block decomposition, DP) and highlight the computational boundaries and potential for extending PMMS to broader graph classes. The work advances local fairness in graph-based allocation settings and opens questions about PMMS existence in general graphs and for other local fairness variants, with practical implications for spatial and temporal resource division.

Abstract

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.

Figures (4)

• Figure 1: A partition $(X(B,c),Y(B,c))$ of a graph.
• Figure 2: Merge operation applied to a block $B^*$.
• Figure 3: Three possible layouts of PMMS partitions for 3 agents on a path.
• Figure 4: Visualisation of all 15 permutations of cut orders in \ref{['thr:three-agents-path']} and the 7 cases that handle them.

Theorems & Definitions (51)

• definition 1: Pairwise MMS
• proposition 1
• proposition 2
• proposition 3
• proposition 4
• proposition 5
• corollary 1
• proposition 6
• theorem 1
• corollary 2