# Envy-free Matchings in Bipartite Graphs and their Applications to Fair Division

arXiv: https://arxiv.org/abs/1901.09527 · HTML: https://mlsanity.com/paper/1901.09527/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2019.

## Built on by (most cited first, 4 of 4)

- [Two-Sided Matching Meets Fair Division](https://mlsanity.com/paper/2107.07404/) (2021)
- [Fairness in Repeated Matching: A Maximin Perspective](https://mlsanity.com/paper/2510.04624/) (2025)
- [Guaranteeing Maximin Shares: Some Agents Left Behind](https://mlsanity.com/paper/2105.09383/) (2021)
- [Graphical House Allocation](https://mlsanity.com/paper/2301.01323/) (2023)

## Abstract (arXiv)

A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share allocation of discrete bads among n agents.