Approximation Algorithms for Not Necessarily Disjoint Clustered TSP - Guttmann-Beck, Nili; Knaan, Eyal; Stern, Michal (Journal of Graph Algorithms and Applications, 2018)
Let $G=(V,E)$ be a complete undirected graph with vertex set $V$, edge set $E$ and let $H = \langle G,\mathcal{S} \rangle $ be a hypergraph, where $\mathcal{S}$ is a set of not necessarily disjoint clusters $S_1,…,S_m$, $S_i \subseteq V$ $\forall i \in \{ 1,…,m \}$. 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.