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 0.3667 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.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio