We consider the problem of guaranteeing a fraction of the maximin-share (MMS) when allocating a set of indivisible items to a set of agents with fractionally subadditive (XOS) valuations. For XOS valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than $1/2$ of the maximin-share to all the agents. Also, a deterministic allocation exists that guarantees $0.219225$ of the maximin-share to each agent. Our results pertain to deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to $3/13 = 0.230769$. We develop new ideas for allocating large items which might be of independent interest. Furthermore, we investigate randomized algorithms and best-of-both-worlds fairness guarantees. We propose a randomized allocation that is $1/4$-MMS ex-ante and $1/8$-MMS ex-post for XOS valuations. Moreover, we prove an upper bound of $3/4$ on the ex-ante guarantee for this class of valuations.

翻译：我们研究在将一组不可分割物品分配给一组具有分数次可加（XOS）估值的代理人时，保证其最大最小份额（MMS）比例的问题。对于XOS估值，已有研究表明：在某些实例中，没有任何分配方案能对所有代理人保证超过$1/2$比例的MMS；同时存在确定性分配方案可保证每位代理人获得$0.219225$比例的MMS。我们的结果涉及确定性与随机分配。在确定性分配方面，我们将分数次可加估值的最优近似保证提升至$3/13 = 0.230769$，并发展出可能具有独立价值的大物品分配新思路。此外，我们研究了随机算法与“两全其美”公平性保证。我们提出一种随机分配方案，对于XOS估值可实现事前$1/4$-MMS与事后$1/8$-MMS保证。进一步地，我们证明对此类估值类别，事前保证的上界为$3/4$。