Approximating Fair Division on D-Claw-Free Graphs

We study the problem of fair allocation of indivisible goods that form a graph and the bundles that are distributed to agents are connected subgraphs of this graph. We focus on the maximin share and the proportional fairness criteria. It is well-known that allocations satisfying these criteria may not exist for many graphs including complete graphs and cycles. Therefore, it is natural to look for approximate allocations, i.e., allocations guaranteeing each agent a certain portion of the value that is satisfactory to her. In this paper we consider the class of graphs of goods which do not contain a star with d+1 edges (where d > 1) as an induced subgraph. For this class of graphs we prove that there is an allocation assigning each agent a connected bundle of value at least 1/d of her maximin share. Moreover, for the same class of graphs of goods, we show a theorem which specifies what fraction of the proportional share can be guaranteed to each agent if the values of single goods for the agents are bounded by a given fraction of this share.