Discrete & Computational Geometry

, Volume 3, Issue 2, pp 177–191 | Cite as

The algebraic degree of geometric optimization problems

  • Chanderjit Bajaj


In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.


Symmetric Group Galois Group Permutation Group Solution Point Galois Theory 
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  1. 1.
    C. Bajaj, Geometric Optimization and Computational Complexity, Computer Science Technical Report TR84-629, Ph.D. thesis, Cornell University, Ithaca, NY, 1984.Google Scholar
  2. 2.
    A. Baker,Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.zbMATHCrossRefGoogle Scholar
  3. 3.
    J. Burns, Abstract definition of groups of degree eight,Amer. J. Math. 37 (1915), 195–214.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Butler and J. McKay, The transitive groups of degree up to 11,Comm. Algebra 11 (1983), 863–911.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    G. E. Collins and R. Loos, Real zeros of polynomials, inComputing Supplementum, vol. 4 (B. Buchbergeret al., eds.), 84–94, Springer-Verlag, Wien, New York, 1982.Google Scholar
  6. 6.
    R. Courant and H. Robbins,What is Mathematics?, Oxford University Press, Oxford, 1941.Google Scholar
  7. 7.
    E. Engeler, Lower bounds by Galois theory,Astérisque 38–39 (1976), 45–52.MathSciNetGoogle Scholar
  8. 8.
    L. Gaal,Classical Galois Theory with Examples, Markham, 1971.Google Scholar
  9. 9.
    M. R. Garey, R. L. Graham, and D. S. Johnson, Some NP-complete geometric problems,Proceedings of the Eighth Symposium on the Theory of Computing, 10–22, 1976.Google Scholar
  10. 10.
    R. L. Graham, Unsolved problem P73, problems and solutions,Bull. EATCS (1984), 205–206.Google Scholar
  11. 11.
    I. N. Herstein,Topics in Algebra, 2nd ed., Wiley, New York, 1975.zbMATHGoogle Scholar
  12. 12.
    D. E. Knuth,The Art of Computer Programming, vol. 2, 2nd edn., Addison-Wesley, Reading, MA, 1981.zbMATHGoogle Scholar
  13. 13.
    H. W. Kuhn, On a pair of dual non-linear programs, inNon-Linear Programming (J. Abadie, ed.), 37–54, North-Holland, Amsterdam, 1967.Google Scholar
  14. 14.
    H. T. Kung, The computational complexity of algebraic numbers,SIAM J. Numer. Anal. 12 (1975), 89–96.MathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Landau and G. L. Miller, Solvability by radicals in polynomial time,Proceedings of the 15th Annual Symposium on the Theory of Computing, 140–151, 1983.Google Scholar
  16. 16.
    J. D. Lipson, Newton's method: a great algebraic algorithm,Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation (SYMSAC), 260–270, 1976.Google Scholar
  17. 17.
    J. McKay, Some remarks on computing Galois groups,SIAM J. Comput. 8 (1979), 344–347.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Z. A. Melzak,Companion to Concrete Mathematics, Wiley, New York, 1973.zbMATHGoogle Scholar
  19. 19.
    G. A. Miller, Memoir on the substitution groups whose degree does not exceed eight,Amer. J. Math. 21 (1899), 287–337.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    A. M. Odlyzko, Personal Communication, May 1985.Google Scholar
  21. 21.
    B. R. Peskin and D. R. Richman, A method to compute minimal polynomials,SIAM J. Algebraic Discrete Methods 6 (1985), 292–299.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    S. M. Rump, Polynomial minimum root separation,Math. Comp. 33 (1979), 327–336.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    J. T. Schwartz, Polynomial Minimum Root Separation (Note to a Paper of S. M. Rump), Robotics Research Technical Report No. 39, New York University, 1985.Google Scholar
  24. 24.
    R. P. Stauduhar, The determination of Galois groups,Math. Comp. 27 (1973), 981–996.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    B. L. Van der Waerden,Modern Algebra, vol. 1, Ungar, New York, 1953.Google Scholar
  26. 26.
    A. Weber,Theory of the Location of Industries (translated by Carl J. Friedrich), The University of Chicago Press, Chicago, 1937.Google Scholar
  27. 27.
    H. Zassenhaus, On the group of an equation,Computers in Algebra and Number Theory, SIAM and AMS Proceedings, 69–88, 1971.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Chanderjit Bajaj
    • 1
  1. 1.Department of Computer SciencePurdue UniversityWest LafayetteUSA

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