3.10.5 Travelling Salesman Problem

What are Hamiltonian paths and cycles?

• A path is a walk in which no vertices are repeated
• A Hamiltonian path is a path in which each vertex in a graph is visited exactly once
• A cycle is a walk that starts and ends at the same vertex and repeats no other vertices
• A Hamiltonian cycle is a cycle which visits each vertex in a graph exactly once
• If a graph contains a Hamiltonian cycle then it is known as a Hamiltonian graph
• A graph is semi-Hamiltonian if it contains a Hamiltonian path but not a Hamiltonian cycle
• The only way to show that a graph is Hamiltonian or semi-Hamiltonian is to find a Hamiltonian cycle or Hamiltonian trail

The travelling salesman problem requires you to find the route of least weight that starts and finishes at the same vertex and visits every other vertex in the graph exactly once.

How do I solve the travelling salesman problem?

• In the classical travelling salesman problem the following conditions are observed:
• the graph is complete
• the direct route between two vertices is the shortest route (it satisfies the triangle inequality)
• List all of the possible Hamiltonian cycles and find the cycle of least weight
• A complete graph with 3 vertices will have 2 possible Hamiltonian cycles, 4 vertices will have 6 possible cycles and a graph with 5 vertices will have 24 possible cycles
• There is no known algorithm that guarantees finding the shortest Hamiltonian cycle in a graph so this method is only suitable for small graphs