Alternative polynomial-time algorithm for Bipartite Matching

TL;DR

This paper introduces a novel polynomial-time algorithm for bipartite matching based on a game-theoretic approach, extending the classical augmenting path method and applying to balanced hypergraphs.

Contribution

It presents an alternative algorithm for bipartite matching using game theory, offering a new perspective and extending to balanced hypergraphs.

Findings

The algorithm correctly finds maximum bipartite matchings.

It generalizes to balanced hypergraphs.

Provides an alternative proof of Hall's theorem.

Abstract

If $G$ is a bipartite graph, Hall's theorem \cite{H35} gives a condition for the existence of a matching of $G$ covering one side of the bipartition. This theorem admits a well-known algorithmic proof involving the repeated search of augmenting paths. We present here an alternative algorithm, using a game-theoretic formulation of the problem. We also show how to extend this formulation to the setting of balanced hypergraphs.