A 5/3 approximation algorithm for the clustered traveling salesman tour and path problems

We consider the ordered cluster traveling salesman problem (OCTSP). In this problem, a vehicle starting and ending at a given depot must visit a set of n points. The points are partitioned into K, K less than or equal to n, prespecified clusters. The vehicle must first visit the points in cluster 1, then the points in cluster 2,..., and finally the points in cluster K so that the distance traveled is minimized. We present a 5/3-approximation algorithm for this problem which runs in O(n(3)) time. We show that our algorithm can also be applied to the path version of the OCTSP: the ordered cluster traveling salesman path problem (OCTSPP). Here the (different) starting and ending points of the vehicle may or may not be prespecified. For this problem, our algorithm is also a 5/3-approximation algorithm. (C) 1999 Elsevier Science B.V. All rights reserved.