# Using a Hamiltonian cycle problem algorithm to assist in solving difficult instances of Traveling Salesman Problem

## TL;DR

This paper presents a hybrid approach combining Hamiltonian cycle algorithms with branch-and-cut methods to solve difficult TSP instances to optimality, including previously unsolved cases.

## Contribution

It introduces a novel hybrid procedure that effectively solves challenging TSP instances by integrating Hamiltonian cycle algorithms with existing solvers.

## Key findings

- Successfully solved a previously unsolved TSP instance from 2002.
- Demonstrated the effectiveness of the hybrid approach on difficult instances.
- Achieved provable optimal solutions for complex TSP problems.

## Abstract

We describe a hybrid procedure for solving the traveling salesman problem (TSP) to provable optimality. We first sparsify the instance, and then use a hybrid algorithm that combines a branch-and-cut TSP solver with a Hamiltonian cycle problem solver. We demonstrate that this procedure enables us to solve difficult instances to optimality, including one which had remained unsolved since its construction in 2002.