The Approximation Ratio of the 2-Opt Heuristic for the Metric Traveling Salesman Problem

The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the well-known Lin-Kernighan algorithm and often used in practice. So far, only upper and lower bounds on the approximation ratio of the 2-Opt heuristic for the metric TSP were known. We prove that for the metric TSP with $n$ cities, the approximation ratio of the 2-Opt heuristic is $\sqrt{n/2}$ and that this bound is tight.