We define and study greedy matchings in vertex-ordered bipartite graphs. It is shown that each vertex-ordered bipartite graph has a unique greedy matching. The proof uses (a weak form of) Newman's lemma. The vertex ordering is called a preference relation. Given a vertex-ordered bipartite graph, the goal is to match every vertex of one vertex class but to leave unmatched as many as possible vertices of low preference in the other concept class. We investigate how well greedy algorithms perform in this setting. It is shown that they have optimal performance provided that the vertex-ordering is cleverly chosen. The study of greedy matchings is motivated by problems in learning theory like illustrating or teaching concepts by means of labeled examples.

翻译：我们在顶点有序的二部图中定义并研究了贪心匹配。证明每个顶点有序的二部图都存在唯一的贪心匹配，该证明使用了纽曼引理（的弱形式）。顶点序被称为偏好关系。给定顶点有序的二部图，目标是匹配某一顶点类中的所有顶点，同时尽可能多地保留另一概念类中偏好较低的顶点不被匹配。我们探讨了贪心算法在此场景下的表现，结果表明，若能巧妙选择顶点顺序，贪心算法具有最优性能。对贪心匹配的研究源于学习理论中的问题，例如通过带标签示例说明或教授概念。