The Fair Division of Hereditary Set Systems

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least $0.3636$ times the maximin share of the agent. This improves upon the current best known guarantee of $0.2$ due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most $0.3738$. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.