Greedy Matchings in Bipartite Graphs with Ordered Vertex Sets

Abstract

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.