We study the problem of fairly dividing indivisible goods among a set of agents under the fairness notion of Any Price Share (APS). APS is known to dominate the widely studied Maximin share (MMS). Since an exact APS allocation may not exist, the focus has traditionally been on the computation of approximate APS allocations. Babaioff et al. studied the problem under additive valuations, and asked (i) how large can the APS value be compared to the MMS value? and (ii) what guarantees can one achieve beyond additive functions. We partly answer these questions by considering valuations beyond additive, namely submodular and XOS functions, with binary marginals. For the submodular functions with binary marginals, also known as matroid rank functions (MRFs), we show that APS is exactly equal to MMS. Consequently, we get that an exact APS allocation exists and can be computed efficiently while maximizing the social welfare. Complementing this result, we show that it is NP-hard to compute the APS value within a factor of 5/6 for submodular valuations with three distinct marginals of {0, 1/2, 1}. We then consider binary XOS functions, which are immediate generalizations of binary submodular functions in the complement free hierarchy. In contrast to the MRFs setting, MMS and APS values are not equal under this case. Nevertheless, we show that under binary XOS valuations, $MMS \leq APS \leq 2 \cdot MMS + 1$. Further, we show that this is almost the tightest bound we can get using MMS, by giving an instance where $APS \geq 2 \cdot MMS$. The upper bound on APS, implies a ~0.1222-approximation for APS under binary XOS valuations. And the lower bound implies the non-existence of better than 0.5-APS even when agents have identical valuations, which is in sharp contrast to the guaranteed existence of exact MMS allocation when agent valuations are identical.

翻译：摘要： 我们研究在任意价格份额(APS)这一公平性概念下，在多个智能体之间公平分配不可分割物品的问题。已知APS优于广泛研究的最大最小份额(MMS)。由于精确的APS分配可能不存在，传统研究重点在于计算近似APS分配。Babaioff等人研究了加性估值下的该问题，并提出了两个问题：(i) APS值相比MMS值能大到何种程度？(ii) 在加性函数之外能实现怎样的保证？我们通过考虑超越加性的估值（即亚模和XOS函数）并针对二元边际情况，部分回答了这些问题。对于具有二元边际的亚模函数（也称为拟阵秩函数，MRFs），我们证明APS恰好等于MMS。因此，我们得到精确APS分配存在且可在最大化社会福利的同时高效计算。作为该结果的补充，我们证明对于具有{0, 1/2, 1}三种不同边际的亚模估值，在5/6因子内计算APS值是NP难的。接着我们考虑二元XOS函数，它是对偶自由层级中二元亚模函数的直接推广。与MRFs设定不同，在此情况下MMS和APS值并不相等。尽管如此，我们证明在二元XOS估值下，$MMS \leq APS \leq 2 \cdot MMS + 1$。此外，通过给出一个$APS \geq 2 \cdot MMS$的实例，我们证明这是利用MMS所能获得的最严格界。APS的上界意味着在二元XOS估值下可实现约0.1222-APS近似比，而下界则表明即便智能体具有相同估值，也不存在优于0.5-APS的分配，这与当智能体估值相同时保证存在精确MMS分配的情况形成鲜明对比。