Approximation algorithms with bounded performance guarantees for the clustered traveling Salesman problem

Let G=(V,E)be a complete undirected graph with vertex set V , edge set E , and edge weights l(e)satisfying triangle inequality. The vertex set Vis partitioned into clustersV1, . . ., Vk . The clustered traveling salesman problemis to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper we consider several variants of this basic problem and provide polynomial time approximation algorithms for them.