Advertisement

Optimization and Approximation Problems Related to Polynomial System Solving

  • Klaus Meer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

We outline some current work in real number complexity theory with a focus on own results. The topics discussed are all located in the area of polynomial system solving. First, we concentrate on a combinatorial optimization problem related to homotopy methods for solving numerically generic polynomial systems. Then, approximation problems are discussed in relation with Probabilistically Checkable Proofs over the real numbers.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S., Lund, C.: Hardness of Approximation. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard problems, pp. 399–446. PWS Publishing, Boston (1996)Google Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Blum, L., Shub, M., Smale, S.: On a Theory of Computation and Complexity over the Real Numbers: NP-completeness, Recursive Functions and Universal Machines. Bull. Amer. Math. Soc. 21, 1–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bürgisser, P., Clausen, M., Shokrollahi, A.: Algebraic Complexity Theory. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bürgisser, P., Cucker, F.: Counting Complexity Classes for Numeric Computations I: Semilinear Sets. SIAM Journal on Computing 33(1), 227–260 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bürgisser, P., Cucker, F.: Counting Complexity Classes for Numeric Computations II: Algebraic and Semialgebraic Sets. Journal of Complexity (to appear)Google Scholar
  10. 10.
    Bürgisser, P., Cucker, F., Lotz, M.: Counting Complexity Classes for Numeric Computations III: Complex Projective Sets. Foundations of Computational Mathematics (to appear)Google Scholar
  11. 11.
    Cox, D., Sturmfels, B. (eds.): Applications of Computational Algebraic Geometry. In: Proc. Sympos. Appl. Math, vol. 53. American Mathematical Society, Providence, RI (1998)Google Scholar
  12. 12.
    Cucker, F., Rojas, J.M. (eds.): Proceedings of the Smalefest 2000. World Scientific, Singapore (2002)zbMATHGoogle Scholar
  13. 13.
    Dedieu, J.P., Malajovich, G., Shub, M.: On the curvature of the central path of linear programming theory. In: Foundations of Computational Mathematics (to appear)Google Scholar
  14. 14.
    Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica 6, 399–436 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liang, H., Bai, F., Shi, L.: Computing the optimal partition of variables in multi-homogeneous homotopy methods. Appl. Math. Comput. 163(2), 825–840 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Li, T., Lin, Z., Bai, F.: Heuristic methods for computing the minimal multi-homogeneous Bézout number. Appl. Math. Comput. 146(1), 237–256 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Malajovich, G., Meer, K.: Computing Multi-Homogeneous Bézout Numbers is Hard. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 244–255. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Meer, K.: Counting Problems over ℝ. Theor. Computer Science 242(1–2), pp. 41–58 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meer, K.: Transparent long proofs: A first PCP theorem for NP. In: Foundations of Computational Mathematics. Springer, Heidelberg (to appear)Google Scholar
  20. 20.
    Meer, K.: On some relations between approximation and PCPs over the real numbers. In: Theory of Computing Systems. Springer, Heidelberg (to appear)Google Scholar
  21. 21.
    Meer, K., Michaux, C.: A Survey on Real Structural Complexity Theory. Bull. Belgian Math. Soc. 4, 113–148 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Renegar, J., Shub, M., Smale, S. (eds.): Proceedings of the AMS Summer Seminar on Mathematics of Numerical Analysis: Real Number Algorithms. Lectures in Applied Mathematics, Park City (1996)Google Scholar
  23. 23.
    Shub, M., Smale, S.: Complexity of Bézout’s Theorem I: Geometric aspects. Journal of the American Mathematical Society 6, 459–501 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Turing, A.M.: On Computable Numbers, with an Application to the Entscheidungsproblem. Proc. London Math. Soc. 42(2), 230–265 (1936)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2001)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Klaus Meer
    • 1
  1. 1.Department of Mathematics and Computer ScienceSyddansk UniversitetOdense MDenmark

Personalised recommendations