Traveling Salesman Must Visit n Cities

In this problem, we are given pairwise distances between n cities and need to determine the visiting sequence such that a salesman can visit all n cities exactly once before returning to the starting city. The objective is to minimize the total travel distance of the salesman's route.

This problem is known as the Traveling Salesman Problem (TSP). TSP is a classic combinatorial optimization problem with broad applications in logistics, manufacturing, electronic design automation, and other fields.

To solve this problem, we can implement various algorithms. A common approach is the greedy nearest-neighbor algorithm, which builds the route by recursively selecting the closest unvisited city from the current location. Another effective method is dynamic programming using the Held-Karp algorithm, which breaks down the problem into overlapping subproblems by storing optimal paths for subsets of cities.

Despite its simple formulation, TSP is computationally challenging as it belongs to the class of NP-hard problems. Therefore, algorithms for solving TSP require significant time and computational resources to find optimal solutions, with exact algorithms having factorial time complexity O(n!) for brute-force approaches.

In summary, the Traveling Salesman Problem represents a significant computational challenge that requires exploring various algorithmic techniques and optimizations to find practical solutions for real-world applications.