Reducing Curse of Dimensionality: Improved PTAS for TSP (with Neighborhoods) in Doubling Metrics

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 given collection of subsets (regions or neighborhoods) in the underlying metric space. We give a randomized polynomial time approximation scheme (PTAS) when the regions are fat weakly disjoint. This notion of regions was first defined when a QPTAS was given for the problem in [SODA 2010: Chan and Elbassioni]. We combine the techniques in the previous work, together with the recent PTAS for TSP [STOC 2012: Bartal, Gottlieb and Krauthgamer] to achieve a PTAS for TSPN. Moreover, more refined procedures are used to improve the dependence of the running time on the doubling dimension k from the previous exp[O(1)k2] (even for just TSP) to exp[O(1)O(k log k)].

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