On the complexity of approximating tsp with neighborhoods and related problems

We prove that various geometric covering problems related to the Traveling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the Group-Traveling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the Group-Steiner-Tree in the Euclidean plane and the Minimum Watchman Tour and Minimum Watchman Path in 3-D. Some inapproximability factors are also shown for special cases of the above problems, where the size of the sets is bounded. Group-TSP and Group-Steiner-Tree where each neighborhood is connected are also considered. It is shown that approximating these variants to within any constant factor smaller than 2 is NP-hard. For the Group-Traveling Salesman and Group-Steiner-Tree Problems in dimension d, we show an inapproximability factor of O(log ^(d-1)/d n) under a plausible conjecture regarding the hardness of Hyper-Graph Vertex-Cover.