Complexity of Locally Fair Allocations on Graphs
Giovanni Varricchione
Abstract:
We study a new model for fair division in which agents, aside from getting allocated a bundle of items (that can include both goods and chores), have also to be assigned a position on a social graph. Hence, natural fairness criteria arise that are not global, as in the classical setting, but local. In particular, we consider three fairness criteria in their local variants: local envy-freeness, local envy-freeness up to one item and local proportionality. For these, we study how complex the task of performing a position assignment (and possibly also an item allocation) which respects a given criterion is. Specifically, we focus on how different graph topologies influence the complexity of these problems. With the exception of local envy-freeness up to one item which, thanks to known facts from the literature, proves to be easily tractable when the item allocation is not fixed, in many cases both tasks prove to be intractable for the criteria we consider. We also provide some parameterized complexity results when only the position assignment has to be done. In particular, we prove tractability, for all criteria but local proportionality, in case the social graph is either a tree or a forest and using two parameters, one of which specifically tailored for these graphs.
Finally, we comment the experimental results we have obtained from randomly generated instances, each with a fixed item allocation done using one of various criteria. The experimentsâ€™ main objective is to measure how different parameters influence the likelihood of instances for which there is a position assignment that respects a given fairness criterion. In particular, local envy-freeness up to one item proves to be the criterion more easily satisfied when the item allocation is random, whereas, in most cases, local proportionality is more likely to be satisfied than local envy-freeness.