We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for $n$ agents with piecewise linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for $\epsilon$-envy-freeness for mixed goods ($\epsilon$-EFM), and present an algorithm that finds an $\epsilon$-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and $1/\epsilon$.

中文翻译：

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混合可分割和不可分割物品的公平分割

我们研究当资源同时包含可分割和不可分割的物品时的公平分割问题。经典的公平概念，例如无嫉妒 (EF) 和无嫉妒最多一种商品 (EF1)，不能直接应用于混合商品设置。在这项工作中，我们为混合商品（EFM）提出了一个新的公平概念，它是 EF 和 EF1 对混合商品设置的直接推广。我们证明对于任意数量的代理总是存在 EFM 分配。我们还提出了有效的算法来计算两个代理和 $n$ 个代理的 EFM 分配，这些代理对可分割的商品具有分段线性估值。最后，我们放宽了无嫉妒的要求，而不是要求混合商品的 $\epsilon$-envy-freeness ($\epsilon$-EFM)，