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Maximum diameter of abstract polytopes

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Pivoting and Extension

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 1))

Abstract

A combinatorial structure called abstract polytope is introduced. It is shown that abstract polytopes are a subclass of pseudo-manifolds and include (combinatorially) simple convex polytopes as a special case.

The main objective is to determine the maximum diameter of abstract polytopes of dimension less than or equal to 5. Those results are relevant to the study of the efficiency of “vertex following” algorithms since the maximum diameter of d-dimensional polytopes with n facets represent, in a sense, the number of iterations required to solve the “worst” problem (with constraint set of d variables with n inequality constraints) using the “best” vertex following algorithm.

Research and reproduction of this report was partially supported by Office of Naval Research, Contract N-00014-67-A-0112-0011; U.S. Atomic Energy Commission, Contract AT[04-3]326 PA #18; National Science Foundation, Grant GP 25738.

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References

  1. I. Adler, G.B. Dantzig and K. Murty, Existence of A-avoiding paths in abstract polytopes, Mathematical Programming Study 1 (1974) 41–42.

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  2. B. Brunbaum, Convex polytopes (Wiley, New York, 1967).

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  3. G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, N.J., 1963).

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  4. V. Klee and D.W. Walkup, The d-step conjecture for polyhedra of dimension d<6, Acta Mathematica 117 (1967) 53–78.

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Author notes

  1. Ilan Adler

    Present address: University of California, Berkeley, Calif., USA

Authors and Affiliations

  1. Stanford University, Stanford, Calif., USA

    Ilan Adler & George B. Dantzig

Authors
  1. Ilan Adler
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  2. George B. Dantzig
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M. L. Balinski

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© 1974 The Mathematical Programming Society

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Adler, I., Dantzig, G.B. (1974). Maximum diameter of abstract polytopes. In: Balinski, M.L. (eds) Pivoting and Extension. Mathematical Programming Studies, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121238

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  • DOI: https://doi.org/10.1007/BFb0121238

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00756-9

  • Online ISBN: 978-3-642-00758-3

  • eBook Packages: Springer Book Archive

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