A note on the traveling salesman reoptimization problem under vertex insertion

We propose a 4 3 -approximation in linear time for the metric traveling salesman reoptimization problem under vertex insertion. This constitutes an improvement of the time complexity of the best known existing bound since in (G. Ausiello, B. Escoffier, J. Monnot and V. Th. Paschos: Reoptimization of minimum and maximum traveling salesman’s tours. J. Discrete Algorithms 7(4): 453-463, 2009) a 4/3-approximation is already given with a complexity O(n). Our algorithm is of independent interest because it does not use a matching to build the approximate solution.