Skip to main content
SpringerLink
Log in

On the Complexity of Approximating TSP with Neighborhoods and Related Problems

  • Conference paper
Algorithms - ESA 2003 (ESA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2832))

Included in the following conference series:

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.

Research supported in part by the Fund for Basic Research Administered by the Israel Academy of Sciences, and a Bikura grant.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arkin, E., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 55(3), 197–218 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Arora, S.: Polynomial-time approximation scheme for Euclidean TSP and other geometric problems. In: Proceedings of the Symposium on Foundations of Computer Science, pp. 2–11 (1996)

    Google Scholar 

  3. Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, Graduate School of Industrial Administration, Carnegy–Mellon University (1976)

    Google Scholar 

  4. Carlsson, S., Jonsson, H., Nilsson, B.J.: Finding the shortest watchman route in a simple polygon. GEOMETRY: Discrete X Computational Geometry 22(3), 377–402 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chin, W., Ntafos, S.: Optimum watchman routes. Information. Processing Letters 28(1), 39–44 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carr, R.D., Vempala, S., Mandler, J.: Towards a 4/3 approximation for the asymmetric traveling salesman problem. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9–11, pp. 116–125. ACM Press, N.Y (2000)

    Google Scholar 

  7. de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. In: ESA: Annual European Symposium on Algorithms, pp. 187–199 (2002)

    Google Scholar 

  8. Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered pcp and the hardness of hypergraph vertex cover. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing, pp. 595–601. ACM Press, New York (2003)

    Chapter  Google Scholar 

  9. Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), January 7-9, pp. 38–46. ACM Press, New York (2001)

    Google Scholar 

  10. Engebretsen, L., Karpinski, M.: Approximation hardness of TSP with bounded metrics. In: ICALP: Annual International Colloquium on Automata, Languages and Programming, pp. 201–212 (2001)

    Google Scholar 

  11. Feige, U.: A threshold of ln n for approximating set cover. JACM: Journal of the ACM 45(4), 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Frieze, A., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric travelling salesman problem. Networks 12, 23–39 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Conference Record of the Eighth Annual ACM Symposium on Theory of Computing, Hershey, Pennsylvania, May 3–5, pp. 10–22 (1976)

    Google Scholar 

  14. Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM Journal on Applied Mathematics 32(4), 835–859 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gudmundsson, J., Levcopoulos, C.: A fast approximation algorithm for TSP with neighborhoods. Nordic Journal of Computing 6(4), 469–488 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Gewali, L., Ntafos, S.C.: Watchman routes in the presence of a pair of convex polygons. Information Sciences 105(1-4), 123–149 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ihler, E.: The complexity of approximating the class Steiner tree problem. In: Schmidt, G., Berghammer, R. (eds.) WG 1991. LNCS, vol. 570, pp. 85–96. Springer, Heidelberg (1992)

    Google Scholar 

  18. Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. JACM 41(5), 960–981 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In: Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, Atlanta, Georgia, January 28–30, pp. 402–408 (1996)

    Google Scholar 

  20. Mitchell, J.S.B.: Geometric Shortest Paths and Network optimization, preliminary edn. Elsevier Science, Amsterdam (2000)

    Google Scholar 

  21. Mata, C., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proceedings of the 11th Annual Symposium on Computational Geometry, pp. 360–369. ACM Press, New York (1995)

    Chapter  Google Scholar 

  22. Nilsson, B.J., Wood, D.: Optimum watchmen in spiral polygons. In: CCCG: Canadian Conference in Computational Geometry, pp. 269–272 (1990)

    Google Scholar 

  23. Papadimitriou, C.H.: Euclidean TSP is NP-complete. Theoretical Computer Science 4, 237–244 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Petrank, E.: The hardness of approximation: Gap location. Computational Complexity 4(2), 133–157 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Papadimitriou, C.H., Vempala, S.: On the approximability of the traveling salesman problem. In: ACM (ed.) Proceedings of the thirty second annual ACM Symposium on Theory of Computing, Portland, Oregon, May 21–23, pp. 126–133. ACM Press, New York (2000) (extended abstract)

    Chapter  Google Scholar 

  26. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 475–484. ACM Press, New York (1997)

    Chapter  Google Scholar 

  27. Slavik, P.: The errand scheduling problem. Technical Report 97-02, SUNY at Buffalo, March 14 (1997)

    Google Scholar 

  28. Xue-Hou, T., Hirata, T., Inagaki, Y.: An incremental algorithm for constructing shortest watchman routes. International Journal of Computational Geometry and Applications 3(4), 351–365 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, 69978, Israel

    Shmuel Safra & Oded Schwartz

Authors
  1. Shmuel Safra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oded Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università degli Studi “Roma Tre”, via della Vasca Navale 79, 00146, Rome, Italy

    Giuseppe Di Battista

  2. School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Uri Zwick

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Safra, S., Schwartz, O. (2003). On the Complexity of Approximating TSP with Neighborhoods and Related Problems. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_41

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-39658-1_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20064-2

  • Online ISBN: 978-3-540-39658-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation