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Jumping Doesn’t Help in Abstract Cubes

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

Abstract

We construct a class of abstract objective functions on the cube, such that the algorithm BottomAntipodal takes exponentially many steps to find the maximum. A similar class of abstract objective functions is constructed for the process BottomTop, also requiring exponentially many steps.

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References

  1. Adler, I., Dantzig, G.B.: Maximum diameter of abstract polytopes. Math. Programming Studies 1, 20–40 (1974)

    MathSciNet  Google Scholar 

  2. Adler, L., Saigal, R.: Long monotone paths in abstract polytopes. Math. Oper. Res. 1, 89–95 (1976)

    MATH  MathSciNet  Google Scholar 

  3. Amenta, N., Ziegler, G.M.: Shadows and slices of polytopes. In: Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 10–19 (1996)

    Google Scholar 

  4. Gärtner, B., Henk, M., Ziegler, G.M.: Randomized simplex algorithms on Klee-Minty cubes. Combinatorica 18(3), 349–372 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gärtner, B., Schurr, I.: Linear Programming and Geometric Unique Sink Orientations. Manuscript in preparation (2004)

    Google Scholar 

  6. Goldfarb, D.: On the complexity of the simplex algorithm. In: Advances in optimization and numerical analysis, pp. 25–38. Kluwer, Dordrecht (1994)

    Google Scholar 

  7. Kaibel, V.: The La Claustra lecture

    Google Scholar 

  8. Kalai, G.: A simple way to tell a simple polytope from its graph. J. Combin. Theory, Ser. A 49(2), 381–383 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kalai, G.: A subexponential randomized simplex algorithm. In: Proc. 24th Annu. ACM Sympos. Theory Comput., pp. 475–482 (1992)

    Google Scholar 

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

    Google Scholar 

  11. Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Morris, W.D.: Randomized principal pivot algorithms for P-matrix linear complementarity problems. Mathematical Programming Ser. A 92, 285–296 (2002)

    Article  Google Scholar 

  13. Schurr, I., Szabó, T.: Finding the sink takes some time, Discrete and Computational Geometry (to appear); An extended abstract appeared in the proceedings of the European Symposium on Algorithms, ESA (2002)

    Google Scholar 

  14. Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proc. 42nd IEEE Symp. on Foundations of Comput. Sci., pp. 547–555 (2001)

    Google Scholar 

  15. Williamson Hoke, K.: Completely unimodal numberings of a simple polytope. Discrete Appl. Math. 20, 69–81 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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© 2005 Springer-Verlag Berlin Heidelberg

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Schurr, I., Szabó, T. (2005). Jumping Doesn’t Help in Abstract Cubes. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_17

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  • DOI: https://doi.org/10.1007/11496915_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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