Connected Fair Allocation of Indivisible Goods
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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 3/4 of the maximin share, while for the remaining graphs the guarantee is at most 1/2. 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.
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