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 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 $q=\infty$.