Approximation algorithms for not necessarily disjoint clustered TSP

Let G = (V, E) be a complete undirected graph with vertex set V, edge set E and let H =< G, S > be a hypergraph, where S is a set of not necessarily disjoint clusters S₁, …, S_m, S_i ⊆ V ∀i ∈ {1, …, m}. The clustered traveling salesman problem CTSP is to compute a shortest Hamiltonian path that visits each one of the vertices once, such that the vertices of each cluster are visited consecutively. In this paper, we present a 4-approximation algorithm for the general case. When the intersection graph is a path, we present a 5/3-approximation algorithm. When the clusters’ sizes are all bounded by a constant and the intersection graph is connected, we present an optimal polynomial time algorithm.