TY - JOUR
T1 - On the complexity of approximating tsp with neighborhoods and related problems
AU - Safra, Shmuel
AU - Schwartz, Oded
N1 - Funding Information:
This research was supported by the Israeli Science Foundation (grant no. 230/02).
PY - 2006/3
Y1 - 2006/3
N2 - 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.
AB - 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.
KW - Approximation
KW - Hardness of approximation
KW - Inapproximability
KW - NP-optimization problems
KW - TSP
KW - TSP with neighborhoods
UR - http://www.scopus.com/inward/record.url?scp=33644891322&partnerID=8YFLogxK
U2 - 10.1007/s00037-005-0200-3
DO - 10.1007/s00037-005-0200-3
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:33644891322
SN - 1016-3328
VL - 14
SP - 281
EP - 307
JO - Computational Complexity
JF - Computational Complexity
IS - 4
ER -