The existence of $\textsf{EFX}$ allocations stands as one of the main challenges in discrete fair division.In this paper, we present symmetrical results on the existence of $\textsf{EFX}$ and its approximate variations for two distinct valuations: restricted additive valuations and $(p,q)$-bounded valuations introduced by Christodoulou \etal \cite{christodoulou2023fair}. In a $(p,q)$-bounded instance, each good has relevance for at most $p$ agents, and any pair of agents shares at most $q$ common relevant goods. We show that instances with $(\infty,1)$-bounded valuations admit $\textsf{EF2X}$ allocations and $\textsf{EFX}$ allocations with at most $\lfloor {n}/{2} \rfloor - 1$ discarded goods, mirroring results for the restricted additive setting \cite{akrami2022ef2x}. We also present ${({\sqrt{2}}/{2})\textsf{-EFX}}$ algorithms for both restricted additive and $(\infty,1)$-bounded subadditive settings. The symmetry of these results suggests these valuations share symmetric structures. Building on this, we propose an $\textsf{EFX}$ allocation for restricted additive valuations when $p=2$ and $q=\infty$. To achieve these results, we further develop the rank concept introduced by Farhadi \etal \cite{farhadi2021almost} and introduce several new concepts such as virtual value, rankpath, and root, which advance the overall understanding of $\textsf{EFX}$ allocations. In addition, we suggest an updating rule based on the virtual values which we believe will lead to broader and more generalized results on $\textsf{EFX}$.

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。本文针对两种不同的估值——受限可加估值以及由Christodoulou等人提出的$(p,q)$-有界估值——给出了关于$\textsf{EFX}$及其近似变体存在性的对称性结果。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，且任意一对智能体至多共享$q$件共同的相关物品。我们证明，具有$(\infty,1)$-有界估值的实例存在$\textsf{EF2X}$分配以及最多丢弃$\lfloor {n}/{2} \rfloor - 1$件物品的$\textsf{EFX}$分配，这与受限可加设置下的结果相呼应。我们还针对受限可加和$(\infty,1)$-有界次可加两种设置，分别提出了${({\sqrt{2}}/{2})\textsf{-EFX}}$算法。这些结果的对称性表明这些估值具有对称的结构。基于此，我们针对$p=2$且$q=\infty$时的受限可加估值，提出了一种$\textsf{EFX}$分配方案。为实现这些结果，我们进一步发展了Farhadi等人提出的秩概念，并引入了虚拟价值、秩路径和根等若干新概念，从而推进了对$\textsf{EFX}$分配的整体理解。此外，我们提出了一种基于虚拟价值的更新规则，我们相信这将有助于获得关于$\textsf{EFX}$的更广泛和更一般化的结果。