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 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 (9 (log n). (C) 2015 Elsevier B.V. All rights reserved.