Fair Division of a Graph

Abstract

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents-or, less restrictively, the number of agent types-is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.

Maximin Share Allocations on Cycles

“…Corollary 2.2 states that for two agents and for any connected graph of goods an mms-allocation exists. On the other hand, Bouveret et al [2] gave an example of nonexistence of an mms-allocation for a cycle and four agents. In fact, as we show in Figure 2, even for three agents it may be that mms-allocations of goods on a cycle do not exist.…”

“…Theorem 3.4 (Bouveret et al [2]). There is a polynomial time algorithm that computes mms (n) (T, u) and a corresponding mms-split given a tree of goods T , a non-negative rational utility function u on the nodes (goods) of T , and an ingeter n ≥ 1.…”

“…In two cases, when m ≤ 2n and when agents can be grouped into a fixed number of types (agents are of the same type if they have the same utility function), this allows us to design polynomial time algorithms for computing maximin share allocations of m goods (on cycles) to n agents or determining that such allocations do not exist. 2. We show that deciding the existence of maximin share allocations of goods on an arbitrary graph is in the class ∆ P 2 .…”

Fairly Allocating Contiguous Blocks of Indivisible Items

“…Recently, Bouveret et al [7] studied the allocation of indivisible items on a line with the contiguity condition and showed that determining whether a contiguous fair allocation exists is NP-hard when the fairness notion considered is either proportionality or envy-freeness. They also considered a more general model of the relationship between items where the items are vertices of an undirected graph.…”

FairRec: Two-Sided Fairness for Personalized Recommendations in Two-Sided Platforms

“…The classical notions of fairness for this problem are envy-freeness (EF) [26,48,50] and proportional fair share (PFS) [47]. Recent literature on practical applications of fair allocation [15,23] have focused on the problem of allocating indivisible goods in budgeted course allocation [16], balanced graph partition [13], or allocation of cardinality constrained group of resources [11]. In such instances, no feasible allocation may satisfy EF or PFS fairness guarantees.…”