We consider the problem of fairly allocating a combination of divisible and indivisible goods. While fairness criteria like envy-freeness (EF) and proportionality (PROP) can always be achieved for divisible goods, only their relaxed versions, such as the ''up to one'' relaxations EF1 and PROP1, can be satisfied when the goods are indivisible. The ''up to one'' relaxations require the fairness conditions to be satisfied provided that one good can be completely eliminated or added in the comparison. In this work, we bridge the gap between the two extremes and propose ''up to a fraction'' relaxations for the allocation of mixed divisible and indivisible goods. The fraction is determined based on the proportion of indivisible goods, which we call the indivisibility ratio. The new concepts also introduce asymmetric conditions that are customized for individuals with varying indivisibility ratios. We provide both upper and lower bounds on the fractions of the modified item in order to satisfy the fairness criterion. Our results are tight up to a constant for EF and asymptotically tight for PROP.

翻译：我们研究公平分配可分割与不可分割商品组合的问题。尽管可分割商品总能实现无嫉妒（EF）和比例性（PROP）等公平标准，但当商品不可分割时，只能实现其松弛版本（例如“至多一个”松弛条件EF1和PROP1）。这类“至多一个”松弛条件要求：在比较中允许完全移除或添加一个商品时，公平条件仍成立。本研究填补了两种极端情形之间的空白，提出适用于混合可分割与不可分割商品分配的“至多一个分数”松弛条件。该分数基于不可分割商品的比例（我们称之为不可分性比率）确定。新概念还引入了针对具有不同不可分性比率的个体定制的非对称条件。我们给出了满足公平准则所需修正商品分数的上界和下界。对于无嫉妒标准，我们的结果在常数范围内是紧致的；对于比例性标准，结果在渐近意义上是紧致的。