We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets (regions or neighborhoods) in the underlying metric space. We give a QPTAS when the regions are what we call α-fat weakly disjoint. This notion combines the existing notions of diameter variation, fatness and disjointness for geometric objects and generalizes these notions to any arbitrary metric space. Intuitively, the regions can be grouped into a bounded number of types, where in each type, the regions have similar upper bounds for their diameters, and each such region can designate a point such that these points are far away from one another. Our result generalizes the PTAS for TSPN on the Euclidean plane by Mitchell and the QPTAS for TSP on doubling metrics by Talwar. We also observe that our techniques directly extend to a QPTAS for the Group Steiner Tree Problem on doubling metrics, with the same assumption on the groups.
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