Examples of Traveling Salesman Problem: Example of Solving with Graphs

Explore practical examples of the Traveling Salesman Problem using graph theory.
By Jamie

Understanding the Traveling Salesman Problem with Graphs

The Traveling Salesman Problem (TSP) is a classic algorithmic problem in the fields of computer science and operations research. It focuses on optimization, specifically the challenge of finding the shortest possible route that visits a set of cities and returns to the origin city. Graph theory serves as a powerful tool in solving TSP, as it allows cities to be represented as vertices and routes as edges. In this article, we will explore three practical examples of the TSP, showcasing how graphs can be utilized to find optimal solutions.

Example 1: Delivery Route Optimization for a Local Bakery

In this scenario, a local bakery needs to deliver goods to five different neighborhoods in a city. The bakery is located at a central point, and the neighborhoods can be represented as nodes in a graph, with the roads between them serving as edges.

To find the most efficient delivery route, the bakery owner can create a weighted graph where the weights represent the distances between neighborhoods. The nodes would be as follows:

  • Node A: Bakery (start point)
  • Node B: Neighborhood 1
  • Node C: Neighborhood 2
  • Node D: Neighborhood 3
  • Node E: Neighborhood 4
  • Node F: Neighborhood 5

The distances (weights) between these nodes are:

  • A to B: 5 miles
  • A to C: 10 miles
  • A to D: 15 miles
  • A to E: 20 miles
  • A to F: 25 miles
  • B to C: 10 miles
  • B to D: 5 miles
  • C to D: 10 miles
  • C to E: 15 miles
  • D to E: 5 miles
  • D to F: 20 miles
  • E to F: 10 miles

Using algorithms such as the nearest neighbor or dynamic programming, the bakery owner can find the optimal route to minimize travel distance and time. For example, a solution could be A → B → D → C → E → F → A, which totals 45 miles.

Notes:

  • Variations of the problem can include time windows for deliveries, vehicle capacity, or the addition of more neighborhoods.

Example 2: School Bus Route Planning

A school district is tasked with planning the most efficient route for a bus that needs to pick up students from various locations. The bus starts at the school (Node A) and must visit five different homes (Nodes B through F) before returning to the school.

The distances between the school and the homes are as follows:

  • A to B: 3 miles
  • A to C: 4 miles
  • A to D: 2 miles
  • A to E: 5 miles
  • A to F: 6 miles
  • B to C: 2 miles
  • B to D: 3 miles
  • C to D: 4 miles
  • C to E: 3 miles
  • D to E: 2 miles
  • D to F: 5 miles
  • E to F: 3 miles

By constructing a graph and applying the brute force method or heuristic approaches, the school district can determine the optimal route. For instance, a potential optimal route could be A → D → B → C → E → F → A, with a total distance of 30 miles.

Notes:

  • Additional considerations might include traffic patterns or student safety, which could affect route selection.

Example 3: Sales Strategy for a Traveling Salesperson

A salesperson needs to visit six clients located in different cities. The objective is to minimize the travel time while ensuring all clients are visited. Each city can be represented as a vertex in a graph, and the edges represent the travel times between them.

Here are the cities and travel times:

  • Node A: Starting point (Home Office)
  • Node B: Client 1
  • Node C: Client 2
  • Node D: Client 3
  • Node E: Client 4
  • Node F: Client 5
  • Node G: Client 6

The travel times between the nodes are:

  • A to B: 1 hour
  • A to C: 2 hours
  • A to D: 3 hours
  • A to E: 4 hours
  • A to F: 5 hours
  • A to G: 6 hours
  • B to C: 1 hour
  • B to D: 2 hours
  • C to D: 1 hour
  • C to E: 2 hours
  • D to E: 1 hour
  • D to F: 2 hours
  • E to G: 1 hour

By using algorithms like branch and bound or genetic algorithms, the salesperson can find the most efficient route. An example of an optimized route could be A → B → C → D → E → F → G → A, totaling approximately 15 hours.

Notes:

  • Variations may include prioritizing certain clients or accommodating unexpected delays, which may alter the route.