Almost Envy-free Allocation of Indivisible Goods: A Tale of Two Valuations

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 \(\textsf{EFX}\).