## On the complexity of approximating TSP with neighborhoods and related problems

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Venue: | Computational Complexity |

Citations: | 29 - 2 self |

### BibTeX

@ARTICLE{Safra_onthe,

author = {S. Safra and O. Schwartz},

title = {On the complexity of approximating TSP with neighborhoods and related problems},

journal = {Computational Complexity},

year = {},

pages = {2006}

}

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### Abstract

We prove that various geometric covering problems, related to the Travelling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the Group-Travelling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the Group-Steiner-Tree in the Euclidean plane and the Minimum Watchman Tour and the Minimum Watchman Path in 3-D. It resolves three open problems presented in the comprehensive survey of Mitchell [Mit00], improves a previously known approximation hardness factor of 2041 2040 [GL00, dBGK+ 02] for the first problem, and it is the first approximation hardness factor for the other problems. 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 neighbourhood 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-Travelling Salesman and Group-Steiner-Tree Problems in dimension d, we show an innapproximability factor of O(log d−1 d Hyper-Graph Vertex-Cover.