Traveling Salesman Problem: Introduction and Approximation Methods

This lecture introduces the Traveling Salesman Problem as a discrete optimization problem, exploring the challenges of finding the shortest path to visit N cities. It delves into the factorial complexity of the problem and discusses the NP class of problems, highlighting the limitations of exact solutions. The instructor explains approximation methods, focusing on the Metropolis algorithm using Markov chains to find near-optimal solutions. The lecture covers the concept of Markov chains, their homogeneity, and transition probabilities between states. It also discusses stationary and limiting distributions in Markov chains, showcasing how these distributions can be used to predict long-term behavior in the context of optimization problems.