Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 281-307 |
| Number of pages | 27 |
| Journal | Computational Complexity |
| Volume | 14 |
| Issue number | 4 |
| DOIs | |
| State | Published - Mar 2006 |
| Externally published | Yes |
Bibliographical note
Funding Information:This research was supported by the Israeli Science Foundation (grant no. 230/02).
Keywords
- Approximation
- Hardness of approximation
- Inapproximability
- NP-optimization problems
- TSP
- TSP with neighborhoods