Skip to main content
SpringerLink
Log in
Book cover

International Conference on Foundations of Software Technology and Theoretical Computer Science

FSTTCS 1985: Foundations of Software Technology and Theoretical Computer Science pp 176–195Cite as

Geometric optimization and the polynomial hierarchy

  • Conference paper
  • First Online:

  • 145 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 206))

Abstract

We illustrate two different techniques of accurately classifying geometric optimization problems in the polynomial hierarchy. We show that if NP≠Co-NP then there are interesting natural geometric optimization problems (location-allocation problems under minsum) in △ P2 that are in neither NP nor Co-NP. Hence, all these problems are shown to belong properly to △ P2 , the second level of the polynomial hierarchy. We also show that if NP≠Co-NP then there are again some interesting geometric optimization problems (location-allocation problems under minmax), properly in △ P2 and furthermore they are complete for a class DP (which is contained in △ P2 and contains NP Co-NP).

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5. References

  1. Aho, A., Hopcroft, J. and Ullman, J., The Design and Analysis of Computer Algorithms, Addison Wesley, 1974.

    Google Scholar 

  2. Bajaj, C., Geometric Optimization and Computational Complexity, Computer Science Tech. Report, Cornell University, Ph.D. Thesis, TR84-629, 1984.

    Google Scholar 

  3. Bajaj, C. and Lim, M., On the duality of intersections and closest points, Computer Science Tech. report, Cornell University, TR83-568, 1983.

    Google Scholar 

  4. Beckenbach, E. and Bellman, R., An Introduction to Inequalities, Random House, 1961.

    Google Scholar 

  5. Cooper, L. Location-Allocation Problems, Operations Research Vol. 11, No. 3, pp 331–343, 1963.

    Google Scholar 

  6. Fowler, R., Paterson, M. and Tanimoto, S., Optimal packing and covering in the plane are NP-complete, Info. Proc. Letters, Vol. 12, no. 2, 1981.

    Google Scholar 

  7. Francis, R. and White J., Facility layout and location — an analytic approach, Prentice Hall, NJ, USA 1974.

    Google Scholar 

  8. Garey, M., Graham, R. and Johnson, D., Some NP-complete geometric problems, Proc. 8th STOC, p. 10–27, 1976.

    Google Scholar 

  9. Garey, M. and Johnson, D., Computers and Intractability: a guide to NP-completeness, Freeman, San Francisco, 1979.

    Google Scholar 

  10. Hadwiger, H. and Debrunner, H., Combinatorial geometry in the plane, (Translated by Klee, V.), Holt, Rhinehart & Winston, 1964.

    Google Scholar 

  11. Hansen, P., Perreur, J. and Thisse, J., Location Theory, Dominance and Convexity: Some further results, Opern. Res., vol. 28, p. 1241–1250, 1980.

    Google Scholar 

  12. Krarup, J. and Pruzan, P., Selected families of discrete location problems, Annals of Discrete Math. 5, North Holland, 1979.

    Google Scholar 

  13. Ladner, R., Lynch, N. and Selman, A., Comparison of Polynomial Time Reducibilities Theoretical Computer Sci. 15, p. 279–289, 1981.

    Article  Google Scholar 

  14. Leggett, E. W. and Moore, D. J., Optimization Problems and the Polynomial Hierarchy, Technical Computer Sci. 15, p. 279–289, 1981.

    Article  Google Scholar 

  15. Lozano-Perez, T., Spatial Planning: A Configuration Space Approach, IEEE Trans. on Computers, V. C-32, p. 108–120, 1983.

    Google Scholar 

  16. Megiddo, N. and Supowit, K. J., On the complexity of some common geometric location problems, Siam J. on Computing, vol. 13, no. 1, p. 182–196, Feb. 1984.

    Article  Google Scholar 

  17. Papadimitriou, C., Worst-case and Probabilistic analysis of a Geometric Location Proglem, Siam J. of Computing, vol. 10, no. 3, 1981.

    Google Scholar 

  18. Papadimitriou, C. and Steiglitz, K., Combinatorial Optimization, Algorithms and Complexity, Prentice Hall, 1982.

    Google Scholar 

  19. Papadimitriou, C. and Yannakakis, M., The Complexity of facets (and some facets of complexity), 14th Annual STOC, May 1982, pp. 255–260.

    Google Scholar 

  20. Selman, A. L., Analogues of semirecursive sets and effective reducibilities to the study of NP completeness, Information & Control, Vol. 52, Jan. 1982, pp36–51.

    Google Scholar 

  21. Shamos, M. I., Computational Geometry, Yale University, Ph.D. Thesis, (Univ. Microfilms International), 1978.

    Google Scholar 

  22. Wendell, R. and Hurter, A., Location Theory, Dominance and Convexity, Opern. Res., vol. 21, p. 314–320, 1973.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Purdue University, 47907, West Lafayette, IN

    Chanderjit Bajaj

Authors
  1. Chanderjit Bajaj
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

S. N. Maheshwari

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bajaj, C. (1985). Geometric optimization and the polynomial hierarchy. In: Maheshwari, S.N. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1985. Lecture Notes in Computer Science, vol 206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16042-6_10

Download citation

  • DOI: https://doi.org/10.1007/3-540-16042-6_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16042-7

  • Online ISBN: 978-3-540-39722-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation