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

• Published in: arXiv.org
• Main Authors: Kaviani, Alireza, Seddighin, Masoud, Shahrezaei, AmirMohammad
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.07.2024
• Subjects: Algorithms; Allocations; Symmetry
• ISSN: 2331-8422

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 \(q=\infty\).