Finding Perfect Matchings in Bipartite Hypergraphs

TL;DR

This paper presents an efficient algorithm for finding perfect matchings in bipartite hypergraphs under a stronger version of Haxell's condition, extending classical matching algorithms to hypergraph settings.

Contribution

It introduces a polynomial-time algorithm for hypergraph perfect matchings under a stronger Haxell's condition, generalizing the Hungarian algorithm for bipartite graphs.

Findings

Algorithm guarantees perfect matchings under the specified condition

Extends classical bipartite matching algorithms to hypergraphs

Potential applications in hypergraph combinatorial problems

Abstract

Haxell's condition is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied. We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell's condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g. Set Packing.