A PTAS for TSP with Fat Weakly Disjoint 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. However, several non-trivial technical hurdles need to be overcome for applying the PTAS framework to TSPN. (1) Heuristic to detect sparse instances. In the STOC 2012 paper, a minimum spanning tree heuristic is used to estimate the portion of an optimal tour within some ball. However, for TSPN, it is not known if an optimal tour would use points inside the ball to visit regions that intersect the ball. (2) Partially cut regions in the recursion. After a sparse ball is identified by the heuristic, the PTAS framework for TSP uses dynamic program to solve the instance restricted to the sparse ball, and recursively solve the remaining instance. However, for TSPN, it is an important issue to decide whether each region partially intersecting the sparse ball should be solved in the sparse instance or considered in the remaining instance. Surprisingly we show that both issues can be resolved by conservatively making the ball in question responsible for all intersecting regions. In particular, a sophisticated charging argument is needed to bound the cost of combining tours in the recursion.



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