Travelling salesman problem
Introduction
The Travelling Salesman Problem (TSP) is a classic algorithmic problem in the field of Computer Science and Operations Research which focuses on optimization. In this problem, a salesman is given a list of cities, and must determine the shortest possible route that allows him to visit each city once and return to his original location.
Mathematical Definition
The TSP can be formally defined as an integer programming problem. Given a complete weighted graph where the vertices represent the cities, the edges represent the paths between the cities, and the weights represent the cost or distance of the paths, the problem is to find a Hamiltonian cycle with the least weight. A Hamiltonian cycle is a closed loop on a graph where every node (vertex) is visited exactly once.
Complexity
The TSP is known to be NP-hard, and its decision problem version, the Hamiltonian cycle problem, is NP-complete. This means that the problem is computationally difficult, in the sense that no efficient solutions are known for the worst-case scenarios. In fact, the problem is so difficult that it is often used as a benchmark for new optimization and approximation algorithms.
Exact Algorithms
Despite its complexity, there are exact algorithms that can solve the TSP. These algorithms can guarantee to find the optimal solution, but they can be very slow, especially for large instances. The most famous exact algorithms for the TSP are the Held-Karp algorithm, which is a dynamic programming approach, and the Concorde algorithm, which is a branch-and-bound method.
Heuristic and Approximation Algorithms
Given the difficulty of solving the TSP exactly, much research has been devoted to developing heuristic and approximation algorithms. These algorithms cannot guarantee to find the optimal solution, but they can often find good solutions in a reasonable amount of time. Some of the most well-known heuristic algorithms for the TSP include the Nearest Neighbor heuristic, the Christofides algorithm, and the Simulated Annealing algorithm.
Real-world Applications
The TSP has many real-world applications, especially in logistics and transportation. For example, it can be used to optimize the routes of delivery trucks, or to plan the itinerary of a salesperson. It is also used in manufacturing, for example to determine the order in which a robotic arm should visit a set of points on a circuit board. In addition, the TSP has applications in astronomy, for example to plan the observations of a moving telescope.