Fair Division of a Graph into Compact Bundles

We study the computational complexity of fair division of indivisible items in an enriched model: there is an underlying graph on the set of items. And we have to allocate the items (i.e., the vertices of the graph) to a set of agents in such a way that (a) the allocation is fair (for appropriate notions of fairness) and (b) each agent receives a bundle of items (i.e., a subset of vertices) that induces a subgraph with a specific "nice structure." This model has previously been studied in the literature with the nice structure being a connected subgraph. In this paper, we propose an alternative for connectivity in fair division. We introduce what we call compact graphs, and look for fair allocations in which each agent receives a compact bundle of items. Through compactness, we try to capture the idea that every agent must receive a set of "closely related" items. We prove a host of hardness and tractability results for restricted input settings with respect to fairness concepts such as proportionality, envy-freeness and maximin share guarantee.