On MMS, APS and XOS

2026-05-09

AI Summary

This work addresses the problem of fair allocation of indivisible goods under XOS (fractionally subadditive) valuations, breaking the long-standing barrier of a 1/4-approximation to the Maximin Share (MMS). By establishing a fundamental connection between the AnyPrice Share (APS) and MMS, the authors design a constructive allocation algorithm tailored for homogeneous XOS valuations and extend it to heterogeneous settings. When the number of agents is sufficiently large, the proposed method achieves—for the first time—an approximation guarantee exceeding 11/40 (≈0.275) for α-MMS, which also implies an α-APS guarantee. This significantly improves upon the previous best-known ratio of 4/17 (≈0.235), offering a new theoretical foundation for achieving both efficiency and fairness in XOS environments.

Abstract

We consider allocations of a set of $m$ indivisible goods to $n$ agents of equal entitlements that have valuations from the class XOS. A previous sequence of works showed allocations that obtain an $α$-approximation for the maximin share (MMS), for values of $α$ that gradually approach $\frac{1}{4}$ from below (the currently known ratio is $\frac{4}{17}$). In this work we attempt to obtain ratios better than $\frac{1}{4}$, and manage to do so for sufficiently large $n$. Our methodology is to first investigate the gap between the anyprice share (APS) and the MMS when all agents have the same XOS valuations, for which we design an allocation algorithm and prove that each agent receives at least $α> \frac{11}{40}$ times the APS. Then, we derive inspiration from this algorithm, and modify it so that it applies also when agents have different XOS valuations. Using this modified version, we show that for some sufficiently large $n_0$, there is an $α$-MMS allocation (in fact, an $α$-APS allocation) for every $n \geq n_0$.

Problem

• Maximin Share
• AnyPrice Share
• XOS valuations
• indivisible goods allocation

Innovation

• Maximin Share
• Anyprice Share
• XOS valuations
• fair division
• approximation algorithm