Skip to main content

Geometry of numbers and integer programming

Summary

  • Invited Presentation
  • Conference paper
  • First Online:
STACS 88 (STACS 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 294))

Included in the following conference series:

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Babai: On Lovász lattice reduction and the nearest lattice point problem. Proc. of STACS '85, Springer Lecture Notes in Computer Science 182 (1985), 13–20.

    Google Scholar 

  2. G. Bergman: Notes on Ferguson and Forcade's generalized euclidean algorithm. Preprint, Dept. of Mathematics, Univ. of California, Berkeley (1980).

    Google Scholar 

  3. L. Bernstein: The Jacobi-Perron algorithm. Springer Lecture Notes in Mathematics 207 (1971).

    Google Scholar 

  4. A. J. Brentjes: Multidimensional continued fraction algorithms. Math. Centre Tracts 155, Amsterdam (1982).

    Google Scholar 

  5. J. W. S. Cassels: An Introduction to the Geometry of Numbers. 2nd edition, Springer, Berlin (1971).

    Google Scholar 

  6. H. Ferguson and R. Forcade: Generalization of the euclidean algorithm for real numbers to all dimensions higher than two. Bull. Amer. Math. Soc. 1, 6 (1979), 912–914.

    Google Scholar 

  7. Ferguson and R. Forcade: Multidimensional euclidean algorithms. J. Reine Angew. Math. 334 (1982), 171–181.

    Google Scholar 

  8. A. M. Frieze, J. Hastad, R. Kannan, J. Lagarias and A. Shamir: Linear congruential generators do not produce random sequences. Siam. J. Computing (to appear).

    Google Scholar 

  9. C. F. Gauss: Disquisitiones Arithmeticae. Leipzig (1801). Deutsche Übersetzung, Springer, Berlin (1889).

    Google Scholar 

  10. P. M. Gruber and C. G. Lekkerkeker: Geometry of Numbers. 2nd edition, North-Holland, Amsterdam-New York (1987).

    Google Scholar 

  11. J. Hastad, B. Just, J. Lagarias and C. P. Schnorr: Polynomial time algorithms for finding integer relations among real numbers. Preliminary version in Proc. of STACS '86, Springer Lecture Notes in Computer Science 210 (1986), 105–118. Full paper to appear in Siam. J. Comput.

    Google Scholar 

  12. B. Helfrich: Algorithms to construct Minkowski reduced and Hermite reduced lattice bases. Theor. Comp. Sci. 41 (1985), 125–139.

    Google Scholar 

  13. C. G. J. Jacobi: Allgemeine Theorie der Kettenbruchähnlichen Algorithmen. J. Reine Angew. Math. 69 (1868), 29–64.

    Google Scholar 

  14. R. Kannan: Minkowski's convex body theorem and integer programming. Mathematics of Operations Research 12 (1987), 415–440.

    Google Scholar 

  15. A. Korkine and G. Zolotareff: Sur les formes quadratiques. Math. Ann. 6 (1873), 366–398.

    Google Scholar 

  16. J. Lagarias and A. M. Odlyzko: Solving low density subset sum problems. 24th Ann. Symp. of Foundation of Comp. Sci. (1983), 1–10.

    Google Scholar 

  17. J. Lagarias, H. W. Lenstra, Jr., and C. P. Schnorr: Korkine-Zolotareff bases and successive minima of a lattice and its reciprocal lattice. To appear in Combinatorica (1987).

    Google Scholar 

  18. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász: Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 513–534.

    Google Scholar 

  19. H. W. Lenstra, Jr.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8 (1983), 538–548.

    Google Scholar 

  20. H. Minkowski: Geometrie der Zahlen. Teubner, Leipzig (1910).

    Google Scholar 

  21. O. Perron: Grundlage für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64 (1907), 1–76.

    Google Scholar 

  22. C. P. Schnorr: A hierarchy of polynomial time lattice basis reduction algorithms. To appear in Theor. Comp. Sci. (1987).

    Google Scholar 

  23. C. P. Schnorr: A more efficient lattice reduction algorithm. To appear in J. Algorithms (1987).

    Google Scholar 

  24. A. Schönhage: Factorization of univariate integer polynomials by diophantine approximation and by an improved basis reduction algorithm. Proc. of 11th ICALP. Springer Lecture Notes in Computer Science 172 (1984), 436–447.

    Google Scholar 

  25. G. Szekeres: Multidimensional continued fraction algorithms. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 13 (1970), 113–140.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Robert Cori Martin Wirsing

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Schnorr, C.P. (1988). Geometry of numbers and integer programming. In: Cori, R., Wirsing, M. (eds) STACS 88. STACS 1988. Lecture Notes in Computer Science, vol 294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035826

Download citation

  • DOI: https://doi.org/10.1007/BFb0035826

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18834-6

  • Online ISBN: 978-3-540-48190-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics