Our work studies the fair allocation of indivisible items to a set of agents, and falls within the scope of establishing improved approximation guarantees. It is well known by now that the classic solution concepts in fair division, such as envy-freeness and proportionality, fail to exist in the presence of indivisible items. Unfortunately, the lack of existence remains unresolved even for some relaxations of envy-freeness, and most notably for the notion of EFX, which has attracted significant attention in the relevant literature. This in turn has motivated the quest for approximation algorithms, resulting in the currently best known $(\phi-1)$-approximation guarantee by Amanatidis et al (2020), where $\phi$ equals the golden ratio. So far, it has been notoriously hard to obtain any further advancements beyond this factor. Our main contribution is that we achieve better approximations, for certain special cases, where the agents agree on their perception of some items in terms of their worth. In particular, we first provide an algorithm with a $2/3$-approximation, when the agents agree on what are the top $n$ items (but not necessarily on their exact ranking), with $n$ being the number of agents. To do so, we also study a general framework that can be of independent interest for obtaining further guarantees.

翻译：我们的工作研究将不可分割物品公平分配给一组智能体的问题，属于建立改进近似保证的范畴。众所周知，公平分配中的经典解概念（如无嫉妒性和比例性）在存在不可分割物品时无法成立。遗憾的是，即使对于无嫉妒性的一些松弛形式（尤其是EFX概念，该概念在相关文献中备受关注），这种缺失仍未得到解决。这进而推动了近似算法的探索，目前已由Amanatidis等人（2020）获得最优的$(\phi-1)$近似保证，其中$\phi$为黄金比例。迄今为止，突破这一因子取得进一步进展一直极为困难。我们的主要贡献在于，针对某些特殊情形（即智能体对部分物品的价值感知达成一致）取得了更好的近似。具体而言，我们首先提出一种$2/3$近似算法，适用于智能体对前$n$个物品达成共识（但不要求精确排序）的情形，其中$n$为智能体数量。为此，我们还研究了一个通用框架，该框架对于获得进一步保证可能具有独立价值。