We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations.

翻译：我们研究在具有加性估值函数的智能体集合中分配不可分割物品的问题，旨在实现针对任意物品的近似无嫉妒性（$\alpha$-EFX）。该问题的最新研究成果包括：（精确）EFX分配在以下情况下存在：（a）智能体数量不超过三个；（b）智能体估值函数最多取两个值；（c）智能体估值函数可通过图表示。对于$\alpha$-EFX，已知对于任意数量具有加性估值函数的智能体，存在$0.618$-EFX分配。本文证明，当满足以下条件时存在$2/3$-EFX分配：（a）智能体数量不超过七个；（b）智能体估值函数最多取三个值；（c）智能体估值函数可通过多重图表示。我们的结果可从两个角度解读：首先，通过将EFX概念松弛为$2/3$-EFX，我们在已知存在精确EFX分配场景的严格推广条件下获得了存在性结果；其次，通过对场景施加限制，我们成功突破了$0.618$的界限，实现了$2/3$的近似保证。因此，本研究推进了近似EFX分配存在性与可计算性的前沿，并为解决精确EFX分配存在性问题提供了新的见解。