# A Linearly-growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem

## TL;DR

This paper presents a linear-size, constructive method to convert the set splitting problem into the directed Hamiltonian cycle problem, maintaining equivalence and enabling solution transfer between the two.

## Contribution

It introduces a novel, linear-size conversion procedure from set splitting to directed Hamiltonian cycle, with proofs of equivalence and solution extraction.

## Key findings

- Conversion preserves problem equivalence
- Input size of converted instance is linear
- Solutions can be transferred between problems

## Abstract

We consider a direct conversion of the, classical, set splitting problem to the directed Hamiltonian cycle problem. A constructive procedure for such a conversion is given, and it is shown that the input size of the converted instance is a linear function of the input size of the original instance. A proof that the two instances are equivalent is given, and a procedure for identifying a solution to the original instance from a solution of the converted instance is also provided. We conclude with two examples of set splitting problem instances, one with solutions and one without, and display the corresponding instances of the directed Hamiltonian cycle problem, along with a solution in the first example.