The existence of $\textsf{EFX}$ allocations stands as one of the main challenges in discrete fair division. In this paper, we present a collection of symmetrical results on the existence of $\textsf{EFX}$ notion and its approximate variations. These results pertain to two seemingly distinct valuation settings: the restricted additive valuations and $(p,q)$-bounded valuations recently introduced by Christodoulou \textit{et al.} \cite{christodoulou2023fair}. In a $(p,q)$-bonuded instance, each good holds relevance (i.e., has a non-zero marginal value) for at most $p$ agents, and any pair of agents share at most $q$ common relevant goods. The only known guarantees on $(p,q)$-bounded valuations is that $(2,1)$-bounded instances always admit $\textsf{EFX}$ allocations (EC'22) \cite{christodoulou2023fair}. Here we show that instances with $(\infty,1)$-bounded valuations always admit $\textsf{EF2X}$ allocations, and $\textsf{EFX}$ allocations with at most $\lfloor {n}/{2} \rfloor - 1$ discarded goods. These results mirror the existing results for the restricted additive setting \cite{akrami2023efx}. Moreover, we present $({\sqrt{2}}/{2})-\textsf{EFX}$ allocation algorithms for both the restricted additive and $(\infty,1)$-bounded settings. The symmetry of these results suggests that these valuations exhibit symmetric structures. Building on this observation, we conjectured that the $(2,\infty)$-bounded and restricted additive setting might admit $\textsf{EFX}$ guarantee. Intriguingly, our investigation confirms this conjecture. We propose a rather complex $\textsf{EFX}$ allocation algorithm for restricted additive valuations when $p=2$ and $q=\infty$.

翻译：$\textsf{EFX}$ 分配的存在性是离散公平分配领域的主要挑战之一。本文针对 $\textsf{EFX}$ 概念及其近似变体的存在性，提出了一系列对称性结果。这些结果涉及两个看似不同的估值设定：受限可加估值以及 Christodoulou 等人近期提出的 $(p,q)$-有界估值 \cite{christodoulou2023fair}。在 $(p,q)$-有界实例中，每件物品至多对 $p$ 个智能体具有相关性（即具有非零边际价值），且任意一对智能体至多共享 $q$ 件共同的相关物品。目前关于 $(p,q)$-有界估值唯一已知的保证是 $(2,1)$-有界实例始终存在 $\textsf{EFX}$ 分配 (EC'22) \cite{christodoulou2023fair}。本文证明，具有 $(\infty,1)$-有界估值的实例始终存在 $\textsf{EF2X}$ 分配，以及最多舍弃 $\lfloor {n}/{2} \rfloor - 1$ 件物品的 $\textsf{EFX}$ 分配。这些结果与受限可加设定下的现有结论 \cite{akrami2023efx} 形成了镜像对应。此外，我们针对受限可加设定与 $(\infty,1)$-有界设定，分别提出了 $({\sqrt{2}}/{2})-\textsf{EFX}$ 分配算法。这些结果的对称性表明，这两类估值具有对称的结构。基于此观察，我们推测 $(2,\infty)$-有界设定与受限可加设定可能具有 $\textsf{EFX}$ 分配保证。有趣的是，我们的研究证实了这一猜想。针对 $p=2$ 且 $q=\infty$ 时的受限可加估值，我们提出了一种较为复杂的 $\textsf{EFX}$ 分配算法。