Second Project: Solving the Travelling Salesman Problem

Overview

The Travelling Salesman Problem (TSP) is a algorithmic problem in the fields of computer science and operations research. It focuses on optimization, where the goal is to determine the shortest possible route that visits a set of given cities and returns to the origin city.

Project Made with:

• João Rebelo
• David Ferreira

Objectives

• Explore different algorithms to solve TSP
• Implement a solution and evaluate its performance
• Project Description

Key Concepts

What is TSP?

The Travelling Salesman Problem can be summarized as follows:

• Input: A list of cities or Edges containing the information on the source/destination, and the distances between each pair of cities.
  Note: There are two types of graphs available for parsing, Toy-Graphs and Extra Fully Connected Graphs, more on this in the Project Description.
• Output: The shortest possible route that visits each city exactly once and returns to the starting point.

Importance of TSP

TSP has applications in various fields such as logistics, planning, and the manufacturing of microchips.

Algorithms we used to Solve TSP

Exact Algorithms

1. BackTracking Approach:
   • The algorithm operates by recursively exploring all possible paths, marking nodes as visited or unvisited as it progresses.
   • For each path, it calculates the total distance traveled and updates the shortest path found whenever a shorter complete tour is identified..
   • Computationally expensive, only feasible for Toy Graphs.

Heuristic Algorithms

1. Triangular Approximation Heuristic:

   • This heuristic involves approximating the solution of a problem by considering triangular inequalities.

2. NearestNeighbour:

   • The best balance of performance and efficiency.
   • It starts from an arbitrary node and repeatedly visits the nearest node until all nodes are visited.

3. K-means Clustering NearestNeighbour:

   • Combines the clustering algorithm with the NearestNeighbour.
   • The graph nodes are divided into a number of clusters.
   • Then the NN heuristic(2.) is applied to each of the clusters.
   • The results are combined to form a complete tour.

4. LinKernighan:

   • This algorithm continues to improve the solution by making local changes until no further improvements can be found.
   • Only feasible in small graphs.