A note on approximation algorithms of the clustered traveling salesman problem

In an earlier paper (Bao and Liu [1]), we considered a version of the clustered traveling salesman problem (CTSP), in which both the starting and ending vertex of each cluster are free to be selected, and proposed a 2.167-approximation algorithm. In this note, we first improve this approximation ratio to 1.9 by introducing a new method to define the inter-node lengths for all the nodes in Step 2 of Algorithm A of Bao and Liu [1]. Based on the above method, we then provide a 2.5-approximation algorithm for another version of CTSP where the starting vertex of each cluster is given while the ending vertex is free to be selected, which improves the previous approximation ratio of 2.643 of Guttmann-Beck et al. [5].