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.

翻译：我们考虑使用最大最小份额度量对不可分割物品进行公平分配的问题。近期该领域的研究重点是将结果扩展到可加估价函数类之外。遵循这一思路，我们研究了物品构成遗传集系统的情况。我们提出一种简单算法，为每个智能体分配一组物品，其价值至少为该智能体最大最小份额的0.3636倍。这改进了目前由Ghodsi等人提出的0.2的最佳已知保证。该算法的分析几乎是最优的：我们给出一个实例表明算法保证值不超过0.3738。我们还证明了在给定每个智能体的估价预言机的情况下，该算法可以在多项式时间内实现。