Combinatorial Linear Programming: Geometry Can Help

  • Bernd Gärtner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1518)


We consider a class A of generalized linear programs on the d-cube (due to Matoušek) and prove that Kalai’s subexponential simplex algorithm Random-Facet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general instances in A. Thus, we identify a “geometric” property of linear programming that goes beyond all abstract notions previously employed in generalized linear programming frameworks, and that can be exploited by the simplex method in a nontrivial setting.


Simplex Method Simplex Algorithm Recursive Call Linear Objective Function Realizable Instance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Adler and R. Saigal. Long monotone paths in abstract polytopes. Math. Operations Research, 1(1):89–95, 1976.MATHMathSciNetGoogle Scholar
  2. 2.
    N. Amenta and G. M. Ziegler. Deformed products and maximal shadows. In J. Chazelle, J.B. Goodman, and R. Pollack, editors, Advances in Discrete and Computational geometry, Contemporary Mathematics. Amer. Math. Soc, 1998.Google Scholar
  3. 3.
    D. Barnette. A short proof of the d-connectedness of d-polytopes. Discrete Math., 137:351–352, 1995.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. G. Bland. New finite pivoting rules for the simplex method. Math. Operations Research, 2:103–107, 1977.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    V. Chvátal. Linear Programming. W. H. Freeman, New York, NY, 1983.MATHGoogle Scholar
  6. 6.
    G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, NJ, 1963.MATHGoogle Scholar
  7. 7.
    B. Gärtner. Randomized Optimization by Simplex-Type Methods. PhD thesis, Freie Universität Berlin, 1995.Google Scholar
  8. 8.
    B. Gärtner. A subexponential algorithm for abstract optimization problems. SIAM J. Comput., 24:1018–1035, 1995.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Gärtner, M. Henk, and G. M. Ziegler. Randomized simplex algorithms on Klee-Minty cubes. Combinatorica (to appear).Google Scholar
  10. 10.
    B. Gärtner and V. Kaibel. Abstract objective function graphs on the 3-cube-a characterization by realizability. Technical Report TR 296, Dept. of Computer Science, ETH Zürich, 1998.Google Scholar
  11. 11.
    B. Gärtner and E. Welzl. Linear programming-randomization and abstract frameworks. In Proc. 13th annu. Symp. on Theoretical Aspects of Computer Science (STAGS), volume 1046 of Lecture Notes in Computer Science, pages 669–687. Springer-Verlag, 1996.Google Scholar
  12. 12.
    D. Goldfarb. On the complexity of the simplex algorithm. In S. Gomez, editor, IV. Workshop on Numerical Analysis and Optimization, Oaxaca, Mexico, 1993.Google Scholar
  13. 13.
    K. Williamson Hoke. Completely unimodal numberings of a simple polytope. Discr. Applied Math., 20:69–81, 1988.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    F. Holt and V. Klee. A proof of the strict monotone 4-step conjecture. In J. Chazelle, J.B. Goodman, and R. Pollack, editors, Advances in Discrete and Computational geometry, Contemporary Mathematics. Amer. Math. Soc, 1998.Google Scholar
  15. 15.
    G. Kalai. A subexponential randomized simplex algorithm. In Proc. 24th annu. ACM Symp. on Theory of Computing., pages 475–482, 1992.Google Scholar
  16. 16.
    G. Kalai. Linear programming, the simplex algorithm and simple polytopes. Math. Programming, 79:217–233, 1997.MathSciNetGoogle Scholar
  17. 17.
    L. G. Khachiyan. Polynomial algorithms in linear programming. U.S.S.R. Comput. Math. and Math. Phys, 20:53–72, 1980.MATHCrossRefGoogle Scholar
  18. 18.
    V. Klee and G. J. Minty. How good is the simplex algorithm? In O. Shisha, editor, Inequalities III, pages 159–175. Academic Press, 1972.Google Scholar
  19. 19.
    J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16:498–516, 1996.MathSciNetMATHGoogle Scholar
  20. 20.
    J. Matoušek. Lower bounds for a subexponential optimization algorithm. Random Structures & Algorithms, 5(4):591–607, 1994.CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 122–132, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Bernd Gärtner
    • 1
  1. 1.Institut für Theoretische InformatikETH Zürich, ETH-ZentrumZürichSwitzerland

Personalised recommendations