Add MST Heuristic for TSP: Problem Generator and Efficient Algorithm

[UditNayak]

Issue: #490

This pull request introduces a solution to the Traveling Salesperson Problem (TSP) using the Minimum Spanning Tree (MST) heuristic. The MST heuristic provides a lower bound estimate for the TSP tour cost, leveraging the MST of a set of cities to approximate the cost of completing a tour.

Key Changes:

1. Heuristic Derivation:

   • Implemented the MST heuristic to estimate TSP tour costs by deriving it from a relaxed version of the TSP where only an MST is required instead of a complete tour.
   • Demonstrated that the MST heuristic is a better approximation compared to the straight-line distance heuristic by considering the network of paths.

2. Problem Generator:

   • Developed a problem generator that creates TSP instances with cities represented by random points within a unit square.
   • This generator allows for testing and benchmarking TSP solutions against randomly generated data.

3. Efficient MST Algorithm:

   • Incorporated Kruskal's algorithm to construct the MST efficiently.
   • Provided a method for using the MST in conjunction with graph search algorithms to solve TSP instances.

Highlights:

• MST Heuristic: Provides a lower bound on the TSP tour cost and dominates simpler heuristics like straight-line distance.
• Problem Generator: Facilitates the creation of diverse TSP instances for testing and validation.
• Algorithm Efficiency: Utilizes an efficient MST construction algorithm to enhance TSP solutions.

Benefits:

• Improved approximation of TSP solutions using the MST heuristic.
• Enhanced ability to test TSP solutions with generated instances.
• Efficient computation of MSTs, aiding in the practical application of the heuristic.

Checklist:

• Implemented MST heuristic for TSP.
• Developed problem generator for TSP instances.
• Incorporated Kruskal's algorithm for MST construction.