Reducing curse of dimensionality: improved PTAS for TSP (with neighborhoods) in doubling metrics

Abstract

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)].