On the Complexity of Approximating TSP with Neighborhoods and Related Problems

Shmuel Safra; Oded Schwartz

doi:10.1007/978-3-540-39658-1_41

On the Complexity of Approximating TSP with Neighborhoods and Related Problems

Shmuel Safra^*, Oded Schwartz

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

15 Scopus citations

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. 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.

Original language	English
Title of host publication	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Editors	Giuseppe di Battista, Uri Zwick
Publisher	Springer Verlag
Pages	446-458
Number of pages	13
ISBN (Print)	3540200649, 9783540200642
DOIs	https://doi.org/10.1007/978-3-540-39658-1_41
State	Published - 2003
Externally published	Yes

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	2832
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

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10.1007/978-3-540-39658-1_41

Cite this

Safra, S., & Schwartz, O. (2003). On the Complexity of Approximating TSP with Neighborhoods and Related Problems. In G. di Battista, & U. Zwick (Eds.), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp. 446-458). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2832). Springer Verlag. https://doi.org/10.1007/978-3-540-39658-1_41

Safra, Shmuel ; Schwartz, Oded. / On the Complexity of Approximating TSP with Neighborhoods and Related Problems. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). editor / Giuseppe di Battista ; Uri Zwick. Springer Verlag, 2003. pp. 446-458 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Safra, S & Schwartz, O 2003, On the Complexity of Approximating TSP with Neighborhoods and Related Problems. in G di Battista & U Zwick (eds), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2832, Springer Verlag, pp. 446-458. https://doi.org/10.1007/978-3-540-39658-1_41

On the Complexity of Approximating TSP with Neighborhoods and Related Problems. / Safra, Shmuel; Schwartz, Oded.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). ed. / Giuseppe di Battista; Uri Zwick. Springer Verlag, 2003. p. 446-458 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2832).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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AB - 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. 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.

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Safra S, Schwartz O. On the Complexity of Approximating TSP with Neighborhoods and Related Problems. In di Battista G, Zwick U, editors, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Springer Verlag. 2003. p. 446-458. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-39658-1_41

On the Complexity of Approximating TSP with Neighborhoods and Related Problems

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