Beating the Logarithmic Barrier for the Subadditive Maximin Share Problem

TL;DR

This paper improves the guaranteed fairness bounds for allocating indivisible goods among subadditive agents, surpassing the previous logarithmic barrier by introducing novel matching and rounding techniques.

Contribution

It presents a new approach that achieves a $1/O(( ext{log log } n)^2)$-MMS guarantee for subadditive agents, improving upon the prior $1/O( ext{log } n ext{ log log } n)$ bound.

Findings

Achieved a new fairness guarantee surpassing the logarithmic barrier.

Developed innovative matching and rounding techniques for subadditive valuations.

Potential applications of these methods in future fair division research.

Abstract

We study the problem of fair allocation of indivisible goods for subadditive agents. While constant-\textsf{MMS} bounds have been given for additive and fractionally subadditive agents, the best existential bound for the case of subadditive agents is $1/O(\log n \log \log n)$. In this work, we improve this bound to a $1/O((\log \log n)^2)$-\textsf{MMS} guarantee. To this end, we introduce new matching techniques and rounding methods for subadditive valuations that we believe are of independent interest and will find their applications in future work.