# From Hall's Marriage Theorem to Boolean Satisfiability and Back

## TL;DR

This paper generalizes Hall's Marriage Theorem to a Fractional Marriage Problem, explores its NP-Completeness, and connects it to Boolean SAT problems, introducing new classes and decision frameworks.

## Contribution

It introduces the Fractional Marriage Problem, shows its NP-Completeness via SAT reduction, and develops methods for recognizing and simplifying these problems within the SAT and matching frameworks.

## Key findings

- Fractional Marriage Problem is NP-Complete.
- Identifies polynomial-time solvable subclasses like the Symmetric Marriage Problem.
- Provides reductions to Tripartite Matching and other problems.

## Abstract

Motivated by the application of Hall's Marriage Theorem in various LP-rounding problems, we introduce a generalization of the classical marriage problem (CMP) that we call the Fractional Marriage Problem. We show that the Fractional Marriage Problem is NP-Complete by reduction from Boolean Satisfiability (SAT). We show that when we view the classical marriage problem (a.k.a. bipartite matching) as a sub-class of SAT we get a new class of polynomial-time satisfiable SAT instances that we call CMP-SAT, different from the classically known polynomial-time satisfiable SAT instances 2-SAT, Horn-SAT and XOR-SAT. We next turn to the problem of recognizing CMP-SAT instances, first using SAT embeddings, and then using their embeddings within the universe of Fractional Marriage Problems (FMPs). In the process we are led to another generalization of the CMP that we call the Symmetric Marriage Problem, which is polynomial time decidable and leads to a slight enlargement of the CMP-SAT class. We develop a framework for simplifying FMP problems to identify CMP instances that we call Fragment Logic. Finally we give a result that sheds light on how expressive the FMP need be to still be NP-Complete. The result gives a second NP-Complete reduction of the FMP, this time to Tripartite Matching. We conclude with a wide assortment of suggested additional problems.