Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Mathematical Foundations of Computer Science

MFCS 2001: Mathematical Foundations of Computer Science 2001 pp 3–17Cite as

On Implications between P-NP-Hypotheses: Decision versus Computation in Algebraic Complexity

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2136))

Abstract

Several models of NP-completeness in an algebraic framework of computation have been proposed in the past, each of them hinging on a fundamental hypothesis of type P≠NP. We first survey some known implications between such hypotheses and then describe attempts to establish further connections. This leads us to the problem of relating the complexity of computational and decisional tasks and naturally raises the question about the connection of the complexity of a polynomial with those of its factors. After reviewing what is known with this respect, we discuss a new result involving a concept of approximative complexity.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Aldaz, J. Heintz, G. Matera, J. L. Montaña, and L. M. Pardo. Time-space tradeoffs in algebraic complexity theory. J. Compl., 16:2–49, 2000.

    Article  MATH  Google Scholar 

  2. A. Alder. Grenzrang und Grenzkomplexität aus algebraischer und topologischer Sicht. PhD thesis, Zürich University, 1984.

    Google Scholar 

  3. A. I. Barvinok. Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor. Random Structures and Algorithms, 14:29–61, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  4. W. Baur. Simplified lower bounds for polynomials with algebraic coefficients. J. Compl., 13:38–41, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Baur and K. Halupczok. On lower bounds for the complexity of polynomials and their multiples. Comp. Compl., 8:309–315, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Berkowitz, C. Rackoff, S. Skyum, and L. Valiant. Fast parallel computation of polynomials using few processors. SIAM J. Comp., 12:641–644, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. Blum, F. Cucker, M. Shub, and S. Smale. Algebraic Settings for the Problem “P ≠ NP?”. In The mathematics of numerical analysis, number 32 in Lectures in Applied Mathematics, pages 125–144. Amer. Math. Soc., 1996.

    Google Scholar 

  8. L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer, 1998.

    Google Scholar 

  9. L. Blum, M. Shub, and S. Smale. On a theory of computation and complexity over the real numbers. Bull. Amer. Math. Soc., 21:1–46, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Bürgisser. The complexity of factors of multivariate polynomials. Preprint, University of Paderborn, 2001, submitted.

    Google Scholar 

  11. P. Bürgisser. On the structure of Valiant’s complexity classes. Discr. Math. Theoret. Comp. Sci., 3:73–94, 1999.

    MATH  Google Scholar 

  12. P. Bürgisser. Completeness and Reduction in Algebraic Complexity Theory, volume 7 of Algorithms and Computation in Mathematics. Springer Verlag, 2000.

    Google Scholar 

  13. P. Bürgisser. Cook’s versus Valiant’s hypothesis. Theoret. Comp. Sci., 235:71–88, 2000.

    Article  MATH  Google Scholar 

  14. P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic Complexity Theory, volume 315 of Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1997.

    Google Scholar 

  15. P. Bürgisser, M. Karpinski, and T. Lickteig. On randomized semialgebraic decision complexity. J. Compl., 9:231–251, 1993.

    Article  MATH  Google Scholar 

  16. P. Bürgisser, T. Lickteig, and M. Shub. Test complexity of generic polynomials. J. Compl., 8:203–215, 1992.

    Article  MATH  Google Scholar 

  17. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comp., 9:251–280, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther. On real Turing machines that toss coins. In Proc. 27th ACM STOC, Las Vegas, pages 335–342, 1995.

    Google Scholar 

  19. H. Fournier and P. Koiran. Are lower bounds easier over the reals? In Proc. 30th ACM STOC, pages 507–513, 1998.

    Google Scholar 

  20. H. Fournier and P. Koiran. Lower bounds are not easier over the reals: Inside PH. In Proc. ICALP 2000, LNCS 1853, pages 832–843, 2000.

    Google Scholar 

  21. J. von zur Gathen. Feasible arithmetic computations: Valiant’s hypothesis. J. Symb. Comp., 4:137–172, 1987.

    Article  MATH  Google Scholar 

  22. B. Griesser. Lower bounds for the approximative complexity. Theoret. Comp. Sci., 46:329–338, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  23. D.Yu. Grigoriev and M. Karpinski. Randomized quadratic lower bound for knapsack. In Proc. 29th ACM STOC, pages 76–85, 1997.

    Google Scholar 

  24. J. Grollmann and A. L. Selman. Complexity measures for public-key cryptosystems. SIAM J. Comp., 17(2):309–335, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Heintz and J. Morgenstern. On the intrinsic complexity of elimination theory. Journal of Complexity, 9:471–498, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. R. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. Electronic Colloquium on Computational Complexity, 2000. Report No. 79.

    Google Scholar 

  27. E. Kaltofen. Single-factor Hensel lifting and its application to the straight-line complexity of certain polynomials. In Proc. 19th ACM STOC, pages 443–452, 1986.

    Google Scholar 

  28. E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Micali, editor, Randomness and Computation, pages 375–412. JAI Press, Greenwich CT, 1989.

    Google Scholar 

  29. P. Koiran. Computing over the reals with addition and order. Theoret. Comp. Sci., 133:35–47, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  30. P. Koiran. A weak version of the Blum, Shub & Smale model. J. Comp. Syst. Sci., 54:177–189, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  31. P. Koiran. Circuits versus trees in algebraic complexity. In Proc. STACS 2000, number 1770 in LNCS, pages 35–52. Springer Verlag, 2000.

    Chapter  Google Scholar 

  32. S. Lang. Fundamentals of Diophantine Geometry. Springer Verlag, 1983.

    Google Scholar 

  33. T. Lickteig. On semialgebraic decision complexity. Technical Report TR-90-052, Int. Comp. Sc. Inst., Berkeley, 1990. Habilitationsschrift, Universität Tübingen.

    Google Scholar 

  34. N. Linial, A. Samorodnitsky, and A. Wigderson. A deterministic polynomial algorithm for matrix scaling and approximate permanents. In Proc. 30th ACM STOC, pages 644–652, 1998.

    Google Scholar 

  35. R. J. Lipton and L. J. Stockmeyer. Evaluation of polynomials with super-preconditioning. J. Comp. Syst. Sci., 16:124–139, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  36. S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106:286–303, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  37. F. Meyer auf der Heide. A polynomial linear search algorithm for the n-dimensional knapsack problem. J. ACM, 31:668–676, 1984.

    Article  MATH  Google Scholar 

  38. F. Meyer auf der Heide. Simulating probabilistic by deterministic algebraic computation trees. Theoret. Comp. Sci., 41:325–330, 1985.

    Article  MATH  Google Scholar 

  39. F. Meyer auf der Heide. Fast algorithms for n-dimensional restrictions of hard problems. J. ACM, 35:740–747, 1988.

    Article  Google Scholar 

  40. B. Poizat. Les Petits Cailloux. Number 3 in Nur Al-Mantiq War-Ma’rifah. Aléas, Lyon, 1995.

    Google Scholar 

  41. A. L. Selman. A survey of one-way functions in complexity theory. Math. Systems Theory, 25:203–221, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. Shub and S. Smale. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “NP ≠ P?”. Duke Math. J., 81:47–54, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  43. V. Strassen. Vermeidung von Divisionen. Crelles J. Reine Angew. Math., 264:184–202, 1973.

    MATH  MathSciNet  Google Scholar 

  44. V. Strassen. Einige Resultate über Berechnungskomplexität. Jahr. Deutsch. Math. Ver., 78:1–8, 1976.

    MATH  MathSciNet  Google Scholar 

  45. L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM STOC, pages 249–261, 1979.

    Google Scholar 

  46. L. G. Valiant. The complexity of computing the permanent. Theoret. Comp. Sci., 8:189–201, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  47. L. G. Valiant. Reducibility by algebraic projections. In Logic and Algorithmic: an International Symposium held in honor of Ernst Specker, volume 30, pages 365–380. Monogr. No. 30 de l’Enseign. Math., 1982.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, University of Paderborn, D-33095, Paderborn, Germany

    Peter Bürgisser

Authors
  1. Peter Bürgisser
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematical Institute, AS CR, Žitná 25, 115 67, Praha 1, Czech Republic

    Jiří Sgall

  2. Faculty of Mathematics and Physics Institute for Theoretical Computer Science (ITI), Charles University, Malostranské náměstí 25, 118 00, Praha 1, Czech Republic

    Aleš Pultr  & Petr Kolman  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bürgisser, P. (2001). On Implications between P-NP-Hypotheses: Decision versus Computation in Algebraic Complexity. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_2

Download citation

  • DOI: https://doi.org/10.1007/3-540-44683-4_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42496-3

  • Online ISBN: 978-3-540-44683-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation