SIAM Journal on Computing, Volume 50, Issue 4, Page 1336-1358, January 2021.
The fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a “fair” manner, where every agent has a valuation for each subset of goods. We assume monotone valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is “envy-freeness up to any good,” EFX, where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an allocation always exists. We show there is always a partition of the set of goods into $n+1$ subsets $(X_1,\ldots,X_n,P)$, where for $i \in [n]$, $X_i$ is the bundle allocated to agent $i$ and the set $P$ is unallocated (or donated to charity) such that we have (1) envy-freeness up to any good, (2) no agent values $P$ higher than her own bundle, and (3) fewer than $n$ goods go to charity, i.e., $|P| < n$ (typically $m \gg n$). Our proof is constructive and leads to a pseudopolynomial time algorithm to find such an allocation. When agents have additive valuations and $|{P}|$ is large (i.e., when $|P|$ is close to $n$), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation that is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. (Very recently and independently, Amanatidis, Ntokos, and Markakis [Theoret. Comput. Sci., 841 (2020), pp. 94--109], also showed the existence of a 4/7-GMMS allocation.)

中文翻译：

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一点点慈善保证几乎没有嫉妒

SIAM Journal on Computing，第 50 卷，第 4 期，第 1336-1358 页，2021 年 1 月。
不可分割的物品的公平分配是一个经过充分研究的问题。这个问题的目标是以“公平”的方式将 $m$ 商品分配给 $n$ 代理，其中每个代理对每个商品子集都有一个估值。我们假设估值是单调的。无嫉妒是研究最广泛的公平概念。然而，当商品不可分割时，并不总是存在无嫉妒分配。我们在此考虑的公平概念是“对任何商品都无嫉妒”，即 EFX，即在从另一个代理的捆绑包中移除任何单个商品后，没有代理会嫉妒另一个代理。不知道这样的分配是否一直存在。我们证明总是有一个商品集被划分为 $n+1$ 个子集 $(X_1,\ldots,X_n,P)$，其中对于 $i\in[n]$，$X_i$ 是分配给代理 $i$ 的捆绑包，并且集合 $P$ 未分配（或捐赠给慈善机构），因此我们具有（1）任何好处都没有嫉妒，（2）没有代理价值 $P$高于她自己的捆绑包，并且 (3) 捐赠给慈善机构的商品少于 $n$，即 $|P| < n$（通常为 $m \gg n$）。我们的证明是有建设性的，并导致了一个伪多项式时间算法来找到这样的分配。当代理具有附加估值并且 $|{P}|$ 很大时（即，当 $|P|$ 接近 $n$ 时），我们的分配也有一个很好的最大最小份额 (MMS) 保证。此外，我们算法的一个次要变体还表明存在 4/7 分组最大共享 (GMMS) 的分配：这是比 MMS 更强的公平概念。这改进了当前已知的近似 GMMS 分配的最佳界限 1/2。（最近独立地，Amanatidis，Ntokos 和 Markakis [理论。计算。Sci., 841 (2020), pp. 94--109]，也表明存在 4/7-GMMS 分配。）