Skip to main content
SpringerLink
Log in

Linear programming — Randomization and abstract frameworks

  • Invited Lecture
  • Conference paper
  • First Online:
Book cover STACS 96 (STACS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1046))

Included in the following conference series:

Abstract

Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this point is a randomized ‘combinatorial’ algorithm which solves a linear program over d variables and n constraints with expected O(d 2n+e O(√d log d)) arithmetic operations. The bound relies on two algorithms by Clarkson, and the subexponential algorithms due to Kalai, and to Matoušek, Sharir & Welzl.

We review some of the recent algorithms with their analyses. We also present abstract frameworks like LP-type problems and abstract optimization problems (due to Gärtner) which allow the application of these algorithms to a number of non-linear optimization problems (like polytope distance and smallest enclosing ball of points).

Supported by a Leibniz Award from the German Research Society (DFG), We 1265/5-1.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adler, I. and Shamir, R.: A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio. Math. Programming 61 (1993) 39–52

    Article  Google Scholar 

  2. Amenta, N.: Helly-type theorems and generalized linear programming. Discrete Comput. Geom. 12 (1994) 241–261

    Google Scholar 

  3. Amenta, N.: Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem. Proc. 10th Annu. ACM Symp. Computational Geometry (1994) 340–347

    Google Scholar 

  4. Borgwardt, K. H.: The Simplex Method. A Probabilistic Analysis. Volume 1 of Algorithms and Combinatorics, Springer-Verlag, Berlin-Heidelberg (1987)

    Google Scholar 

  5. Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimensions. Proc. 4th SIAM-ACM Symp. on Discrete Alg. (1993) 281–290

    Google Scholar 

  6. Chvátal, V.: Linear Programming. W. H. Freeman, New York, NY (1983).

    Google Scholar 

  7. Clarkson, K. L.: Linear programming in \(O(n3^{d^2 } )\)time. Inform. Process. Lett. 22 (1986) 21–24

    Google Scholar 

  8. Clarkson, K.L.: A Las Vegas algorithm for linear and integer programming when the dimension is small. J. ACM 42(2) (1995) 488–499

    Article  Google Scholar 

  9. Dantzig, G. B.: Linear Programming and Extensions. Princeton University Press, Princeton, NJ (1963).

    Google Scholar 

  10. Danzer, Über ein Problem aus der kombinatorischen Geometrie. Arch. Math. 8 (1957) 347–351

    Article  Google Scholar 

  11. Dyer, M. E.: Linear algorithms for two and three-variable linear programs. SIAM J. Comput. 13 (1984) 31–45.

    Article  Google Scholar 

  12. Dyer, M. E.: On a multidimensional search technique and its application to the Euclidean one-center problem. SIAM J. Comput. 15 (1986) 725–738

    Article  Google Scholar 

  13. Dyer, M. E., Frieze, A. M., A randomized algorithm for fixed-dimensional linear programming. Math. Programming 44 (1989) 203–212

    Article  Google Scholar 

  14. Gärtner, B.: A subexponential algorithm for abstract optimization problems. SIAM J. Comput. 24 (1995) 1018–1035

    Article  Google Scholar 

  15. Gärtner, B.: Randomized Optimization by Simplex-Type Methods. PhD thesis, Institute for Computer Science, Free University Berlin (1995)

    Google Scholar 

  16. Goldwasser, M.: A survey of linear programming in randomized subexponential time. ACM-SIGACT News 26(2) (1995) 96–104

    Article  Google Scholar 

  17. Kalai, G.: A subexponential randomized simplex algorithm. Proc. 24th Annu. ACM Symp. Theory of Computing (1992) 475–482

    Google Scholar 

  18. Khachiyan, L. G.: Polynomial algorithms in linear programming. US.S.R. Comput. Math. and Math. Phys. 20 (1980) 53–72

    Google Scholar 

  19. Klee, V., Minty, G. J.: How good is the simplex algorithm? In O. Shisha, editor, Inequalities III, Academic Press (1972) 159–175

    Google Scholar 

  20. Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Proc. 8th Annu. ACM Symp. Computational Geometry (1992) 1–8; Algorithmica, to appear

    Google Scholar 

  21. Matoušek, J.: Lower bounds for a subexponential optimization algorithm. Random Structures & Algorithms 5(4) (1994) 591–607

    Google Scholar 

  22. Matoušek, J.: On geometric optimization with few violated constraints. Proc. 10th Annu. ACM Symp. Computational Geometry (1994) 312–321

    Google Scholar 

  23. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31 (1984) 114–127

    Article  Google Scholar 

  24. Seidel, R.: Small-dimensional linear programming and convex hulls made easy. Discrete Comput. Geom. 6 (1991) 423–434

    Google Scholar 

  25. Shrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    Google Scholar 

  26. Sharir, M., Welzl, E.: A combinatorial bound for linear programming and related problems. Proc. 9th Symp. Theo. Asp. Comp. Sci., Lecture Notes in Computer Science 577 (1992) 569–579

    Google Scholar 

  27. Sharir, M., Welzl, E.: Rectilinear and polygonal p-piercing and p-center problems. Manuscript, submitted (1995)

    Google Scholar 

  28. Sharir, M., Welzl, E.: Circular and spherical separability. Manuscript, submitted (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustr. 9, D-14195, Berlin, Germany

    Bernd Gärtner & Emo Welzl

Authors
  1. Bernd Gärtner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emo Welzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Claude Puech Rüdiger Reischuk

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gärtner, B., Welzl, E. (1996). Linear programming — Randomization and abstract frameworks. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_54

Download citation

  • DOI: https://doi.org/10.1007/3-540-60922-9_54

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60922-3

  • Online ISBN: 978-3-540-49723-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation