On the set of stable matchings of a bipartite graph

TL;DR

This paper surveys key results on the structure, polyhedral aspects, and algorithms related to stable matchings in bipartite graphs, highlighting their theoretical properties and providing concise proofs.

Contribution

It offers a comprehensive overview of known results in stable bipartite matchings, emphasizing structural, polyhedral, and algorithmic insights with simplified proofs.

Findings

Structural properties of stable matchings

Polyhedral characterizations of matching sets

Algorithmic approaches to finding stable matchings

Abstract

The topic of stable matchings (marriages) in a bipartite graph has become widely popular, starting with the appearance of the classical work by Gale and Shapley. We give a detailed survey on selected known results in this field that demonstrate structural, polyhedral and algorithmic properties of such matchings and their sets, providing our description with relatively short proofs.