# On a Generalization of the Marriage Problem

## TL;DR

This paper generalizes the classical marriage problem by introducing the Symmetric Marriage Problem and its non-bipartite variant, extending Hall's and Tutte's theorems to broader matching scenarios with finite and infinite cases.

## Contribution

It introduces the Symmetric Marriage Problem, establishes necessary and sufficient conditions for solutions, and extends classical theorems to new matching problem variants.

## Key findings

- Solution exists iff a variation of Hall's Condition holds
- Finite and infinite versions are proved
- A non-bipartite generalization of Tutte's Theorem is provided

## Abstract

We present a generalization of the marriage problem underlying Hall's famous Marriage Theorem to what we call the Symmetric Marriage Problem, a problem that can be thought of as a special case of Maximal Weighted Bipartite Matching. We show that there is a solution to the Symmetric Marriage Problem if and only if a variation on Hall's Condition holds on each of the bipartitions. We prove both finite and infinite versions of this result and provide applications. We also introduce a non-bipartite version of the problem and show that a generalization of Tutte's Theorem applies.