# Approximation Algorithms for Maximin Fair Division

arXiv: https://arxiv.org/abs/1703.01851 · HTML: https://mlsanity.com/paper/1703.01851/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Mar 2017.

## Built on by (most cited first, 8 of 8)

- [Towards Fair Recommendation in Two-Sided Platforms](https://mlsanity.com/paper/2201.01180/) (2021)
- [Online Fair Division with Additional Information](https://mlsanity.com/paper/2505.24503/) (2025)
- [Guaranteeing Maximin Shares: Some Agents Left Behind](https://mlsanity.com/paper/2105.09383/) (2021)
- [(Almost) Envy-Free, Proportional and Efficient Allocations of an Indivisible Mixed Manna](https://mlsanity.com/paper/2202.02672/) (2022)
- [Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division](https://mlsanity.com/paper/2507.14957/) (2025)
- [On Approximate MMS Allocations on Restricted Graph Classes](https://mlsanity.com/paper/2508.06343/) (2025)
- [Maximin Share Guarantees via Limited Cost-Sensitive Sharing](https://mlsanity.com/paper/2602.20541/) (2026)
- [Improving the Bounds on Approximating Weighted Maximin Share for Chores](https://mlsanity.com/paper/2610.15520/) (2026)

## Abstract (arXiv)

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.