An Improved Kalai-Kleitman Bound for the Diameter of a Polyhedron

@article{Todd2014AnIK,
  title={An Improved Kalai-Kleitman Bound for the Diameter of a Polyhedron},
  author={Michael J. Todd},
  journal={SIAM J. Discret. Math.},
  year={2014},
  volume={28},
  pages={1944-1947}
}
  • M. Todd
  • Published 14 February 2014
  • Mathematics
  • SIAM J. Discret. Math.
Kalai and Kleitman [Bull. Amer. Math. Soc. (N.S.), 26 (1992), pp. 315--316] established the bound $n^{\log(d) + 2}$ for the diameter of a $d$-dimensional polyhedron with $n$ facets. Here we improve the bound slightly to $(n-d)^{\log(d)}$. 
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References

SHOWING 1-10 OF 25 REFERENCES
A counterexample to the Hirsch conjecture
TLDR
This paper presents the rst counterexample to the Hirsch Conjecture, obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch Conjecture, posed in 1957, stated that the graph of a d-dimensional polytope or polyhedron with n facets cannot have diameter greater than n−d. The conjecture itself has been disproved,
An Update on the Hirsch Conjecture
The Hirsch conjecture was posed in 1957 in a question from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than
Lectures on Polytopes
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward
A quasi-polynomial bound for the diameter of graphs of polyhedra
The diameter of the graph of a d-dimensional polyhedron with n facets is at most nlog d+2 Let P be a convex polyhedron. The graph of P denoted by G(P ) is an abstract graph whose vertices are the
Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture
Linear programming, the simplex algorithm and simple polytopes
TLDR
This paper surveys some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes and describes subexponential randomized pivot rules and upper bounds on the diameter of graphs of poly topes.
Paths on polyhedra. II.
Abstract : The paper is motivated in part by a hypothesis of Saaty (Operations research, 12 (1964), p. 159-161) concerning the paths in polyhedra which are produced by variants of the simplex
The d-Step Conjecture and Its Relatives
TLDR
This report summarizes what is known about the d-step conjecture and its relatives and includes the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work.
Maximum Diameter of Abstract Polytopes
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
...
...