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

TL;DR

This paper advances the understanding of envy-free allocations for indivisible goods by establishing new existence results and algorithms for specific valuation classes, including restricted additive and bounded valuations.

Contribution

It introduces novel algorithms and theoretical results for EFX allocations under various valuation restrictions, expanding the scope of fair division solutions.

Findings

Existence of EF2X allocations for (∞,1)-bounded valuations.

EFX allocations with limited discarded goods for certain valuation classes.

New concepts like virtual value and rankpath improve understanding of EFX allocations.

Abstract

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 $\textsf{EFX}$.