The Approximation Ratio of the Greedy Algorithm for the Metric Traveling Salesman Problem

We prove that the approximation ratio of the greedy algorithm for the metric Traveling Salesman Problem is $\mathrm{\Theta }(\log n)$. Moreover, we prove that the same result also holds for graphic, Euclidean, and rectilinear instances of the Traveling Salesman Problem. Finally we show that the approximation ratio of the Clarke-Wright savings heuristic for the metric Traveling Salesman Problem is $\mathrm{\Theta }(\log n)$.