Envy-free matchings in bipartite graphs and their applications to fair division

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-[2n=3j maximin-share allocation of discrete bads among n agents. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

An envy-free matching in a bipartite graph refers to a matching in which no unmatched vertex in X is adjacent to a matched vertex in Y. We prove the existence of a unique partition in every bipartite graph, where all envy-free matchings are contained within one of the partition sets. Based on this structural theorem, we present a polynomial-time algorithm for finding an envy-free matching with maximum cardinality.