Achieving Maximin Share and EFX/EF1 Guarantees Simultaneously
Hannaneh Akrami, Nidhi Rathi
TL;DR
This work studies fair division of indivisible goods with additive valuations, aiming to simultaneously guarantee maximin share ($\mathsf{MMS}$) and envy-based fairness ($\mathsf{EFX}$/\mathsf{EF}1$). It introduces constructive algorithms that achieve a $\frac{2}{3}$-$\mathsf{MMS}$ bound together with $\mathsf{EFX}$ (partial) and $\mathsf{EF}1$ (complete), with computational regimes ranging from pseudo-polynomial to polynomial-time under relaxations like $(1-\delta)$-$\mathsf{EFX}$ or $(1-\delta)$-$\mathsf{EF}1$. The techniques combine partition-based MMS procedures, threshold-graph matchings, and envy-control steps, and incorporate charity-based approaches to bound leftovers, improving over prior $\tfrac{1}{2}$-$\mathsf{MMS}$+EFX results. The results establish a new pathway for balancing envy- and share-based fairness, enabling practical algorithms with provable guarantees in both exact and relaxed settings. These advances have potential impact on fair division in distributed systems and multi-agent resource allocation by offering stronger simultaneous guarantees with tractable computation.
Abstract
We study the problem of computing \emph{fair} divisions of a set of indivisible goods among agents with \emph{additive} valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having \emph{envy-based} or \emph{share-based} lens. For the discrete setting of resource-allocation problems, \emph{envy-free up to any good} (EFX) and \emph{maximin share} (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors [Chaudhury et al., 2021, Amanatidis et al., 2020]. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both $2/3$-MMS and EFX, and (ii) a complete allocation that is both $2/3$-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to $2/3-\varepsilon$ for any constant $\varepsilon > 0$ and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to $(1-δ)$-EFX (or $(1-δ)$-EF1) for any constant $δ>0$. In particular, we improve from the best approximation factor known prior to our work, which computes partial allocations that are $1/2$-MMS and EFX in pseudo-polynomial time [Chaudhury et al., 2021].