Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a $(\varphi -1)$-approximation of envy-freeness up to any good (

) and a $\frac{2}{\varphi +2}$-approximation of groupwise maximin share fairness (

), where ϕ is the golden ratio ($\varphi \approx 1.618$). The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (

) and a 2/3-approximation of pairwise maximin share fairness (

). While

is our primary focus, we also exhibit how to fine-tune our algorithm and further improve the guarantees for

or

.

Finally, we show that

—and thus

and

—allocations always exist when the number of goods does not exceed the number of agents by more than two.

中文翻译：

---

用一颗石头放多只鸟：通过嫉妒循环消除，击败EFX和GMMS的1/2

在过去的十年中，已经采取了几种放宽嫉妒自由的措施，这些措施是针对在不可分割的商品中公平分配而设计的。由于大多数这些概念缺乏普遍存在的结果，因此已经非常重视建立近似保证。在这项工作中，我们提出了一种简单的算法，该算法具有普遍的公平意义，即它一次返回关于四个这样的公平概念具有良好近似保证的分配。特别是，这是第一个实现$（\varphi -\mathrm{1个}）$-羡慕自由的近似值，直到任何好处（

）和一个 $\frac{2}{\varphi +2}$-maximin分组公平性的近似（

），其中φ是黄金比例（$\varphi \approx 1.618$）。在这项工作之前，对于这些公平性概念中的任何一个，在多项式时间内，最著名的近似因子为1/2。此外，返回的分配最多可以使您羡慕不已一种商品（

）和成对的maximin股票公平性的2/3逼近（

）。而

是我们的主要重点，我们还将展示如何微调算法并进一步提高对

要么

。

最后，我们证明

-因此

和

-当商品数量不超过代理商数量的两倍以上时，分配总存在。