We study the fair allocation of indivisible goods under the assumption that the goods form an undirected graph and each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We establish graph-specific maximin share guarantees, which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. Unlike in previous work, our guarantees are with respect to the complete-graph maximin share, which allows us to compare possible guarantees for different graphs. For instance, we show that for biconnected graphs it is possible to obtain at least @math of the maximin share, while for the remaining graphs the guarantee is at most @math . In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretical tools and concepts to fair division problems.
Fair allocation of indivisible goods has received considerable attention from the research community, especially in the last few years. We refer to surveys by @cite_7 , @cite_10 , and @cite_14 for an overview of recent developments in the area.
@cite_14 @cite_10 @cite_7
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Fair division, a key concern in the design of many social institutions, has for 70 years been the subject of interdisciplinary research at the interface of mathematics, economics, and game theory. ... Efficient algorithms are presented for partitioning a graph into connected components, biconnected components and simple paths. The algorithm for partitioning of a graph into simple paths of iterative and each iteration produces a new path between two vertices already on paths. (The start vertex can be specified dynamically.) If V is the number of vertices and E is the number of edges, each algorithm requires time and space proportional to max ( V, E ) when executed on a random access computer.