On Covering Points with Minimum Turns

  • Minghui Jiang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)

Abstract

We point out mistakes in several previous FPT algorithms for k -Link Covering Tour and its variants in ℝ2, and show that the previous NP-hardness proofs for Minimum-Link Rectilinear Covering Tour and Minimum-Link Rectilinear Spanning Path in ℝ3 are incorrect. We then present new NP-hardness proofs for the two problems in ℝ10.

Keywords

Traveling Salesman Problem Computational Geometry Hamiltonian Path Grid Graph Covering Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwal, A., Coppersmith, D., Khanna, S., Motwani, R., Schieber, B.: The angular-metric traveling salesman problem. SIAM Journal on Computing 29, 697–711 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arkin, E.M., Mitchell, J.S.B., Piatko, C.D.: Minimum-link watchman tours. Information Processing Letters 86, 203–207 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal covering tours with turn costs. SIAM Journal on Computing 35, 531–566 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bereg, S., Bose, P., Dumitrescu, A., Hurtado, F., Valtr, P.: Traversing a set of points with a minimum number of turns. Discrete & Computational Geometry 41, 513–532 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Collins, M.J.: Covering a set of points with a minimum number of turns. International Journal of Computational Geometry and Applications 14, 105–114 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Collins, M.J., Moret, B.M.E.: Improved lower bounds for the link length of rectilinear spanning paths in grids. Information Processing Letters 68, 317–319 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Estivill-Castro, V., Heednacram, A., Suraweera, F.: NP-completeness and FPT results for rectilinear covering problems. Journal of Universal Computer Science 15, 622–652 (2010)MathSciNetGoogle Scholar
  8. 8.
    Estivill-Castro, V., Heednacram, A., Suraweera, F.: FPT-algorithms for minimum-bends tours. International Journal of Computational Geometry 21, 189–213 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Fekete, S.P., Woeginger, G.J.: Angle-restricted tours in the plane. Computational Geometry: Theory and Applications 8, 195–218 (1997)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gaur, D.R., Bhattacharya, B.: Covering points by axis parallel lines. In: Proceedings of the 23rd European Workshop on Computational Geometry, pp. 42–45 (2007)Google Scholar
  11. 11.
    Grantson, M., Levcopoulos, C.: Covering a set of points with a minimum number of lines. In: Proceedings of the 22nd European Workshop on Computational Geometry, pp. 145–148 (2006)Google Scholar
  12. 12.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Applied Mathematics 30, 29–42 (1991)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamiltonian paths in grid graphs. SIAM Journal on Computing 11, 676–686 (1982)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Langerman, S., Morin, P.: Covering things with things. Discrete & Computational Geometry 33, 717–729 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kranakis, E., Krizanc, D., Meertens, L.: Link length of rectilinear Hamiltonian tours in grids. Ars Combinatoria 38, 177–192 (1994)MathSciNetMATHGoogle Scholar
  16. 16.
    Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Operations Research Letters 1, 194–197 (1982)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Stein, C., Wagner, D.P.: Approximation Algorithms for the Minimum Bends Traveling Salesman Problem. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 406–421. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Wagner, D.P.: Path Planning Algorithms under the Link-Distance Metric. Ph.D. thesis, Dartmouth College (2006)Google Scholar
  19. 19.
    Wang, J., Li, W., Chen, J.: A parameterized algorithm for the hyperplane-cover problem. Theoretical Computer Science 411, 4005–4009 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Minghui Jiang
    • 1
  1. 1.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations