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

## TL;DR

This paper studies envy-free matchings in bipartite graphs, providing structural insights and polynomial algorithms for maximum envy-free matchings, and applies these concepts to various fair division problems involving both continuous and discrete resources.

## Contribution

It introduces a unique partition theorem for envy-free matchings, along with polynomial algorithms for maximum envy-free matchings and their weighted variants, and demonstrates applications in fair division.

## Key findings

- Existence of a unique partition for envy-free matchings
- Polynomial-time algorithms for maximum envy-free matchings
- Applications to fair division of cakes and discrete goods/bads

## Abstract

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.