Greedy Matchings in Bipartite Graphs with Ordered Vertex Sets
Hans U. Simon
TL;DR
This work studies greedy matchings in vertex-ordered bipartite graphs and proves that every such graph admits a unique greedy matching by modeling matchings as states in a finite abstract rewriting system and applying Newman's lemma. It shows that, with an appropriate ordering of the left vertex set $L$, the greedy matching yields an $L$-saturating solution of minimum order, and extends the framework to matchings with the Preference-Based Teaching (PBT) property. In the learning-theoretic setting of consistency graphs, the largest sample matched by the greedy algorithm is bounded above by $\lceil \log |C| \rceil$, with a matching lower bound establishing near-tightness for the full concept class $C_{all}=2^X$ via a constant $\gamma_0$ determined from $h(\gamma)=(2e/\gamma)^\gamma$. The results connect combinatorial greedy procedures with teaching and illustrating concepts in learning theory, showing how carefully chosen vertex orderings can yield optimal or near-optimal outcomes.
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.