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.

翻译：在双边图中,与X和Y部分的匹配被称为“不折不扣的顶点”* 。 如果X中没有不相配的顶点是Y中匹配的顶点的附近。 每一个完美的匹配都是无嫉妒的, 但即使没有完美的匹配, 也存在无嫉妒的匹配。 我们证明每个双边图有一个独特的分隔点, 使得所有无嫉妒的匹配都包含在分区组合中。 使用这个结构性理论, 我们提供一种多元时间算法, 以寻找一个无嫉妒的顶点匹配。 对于边缘加权的双面图, 我们提供一种多元时间算法, 以寻找一个最大心性、 无嫉妒的最小总重量的无嫉妒匹配。 我们展示如何将无嫉妒的匹配用于各种公平的划分问题, 包括连续资源( 蛋糕 ) 或离散的匹配。 特别是, 我们提出一个比例蛋糕切换式的对称算法, 一种在最大( 2n-2) 之间最大比例分配离心货物的算法, 以及一个最大正位代理方之间最差的正位分配算法 。