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)}\$.
Improvement of Kalai-Kleitman bound for the diameter of a polyhedron
• Mathematics
• 2014
Recently, Todd got a new bound on the diameter of a polyhedron using an analysis due to Kalai and Kleitman in 1992. In this short note, we prove that the bound by Todd can further be improved.
A simple proof of tail--polynomial bounds on the diameter of polyhedra
• Mathematics
• 2016
Let \$\Delta(d,n)\$ denote the maximum diameter of a \$d\$-dimensional polyhedron with \$n\$ facets. In this paper, we propose a unified analysis of a recursive inequality about \$\Delta(d,n)\$ established
An improved upper bound on the diameters of subset partition graphs
• Mathematics
• 2014
In 1992, Kalai and Kleitman proved the first subexponential upper bound for the diameters of convex polyhedra. Eisenbrand et al. proved this bound holds for connected layer families, a novel approach
An Asymptotically Improved Upper Bound on the Diameter of Polyhedra
An asymptotically improved upper bound of (n-d)log2O(d/logd) is shown, which is tight for d-dimensional polyhedron facets and has been improved for \$\$d \ge 3\$\$d≥3 in subsequent studies.
Tail diameter upper bounds for polytopes and polyhedra
• Mathematics
• 2016
In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra.
The diameter of lattice zonotopes
• Mathematics
Proceedings of the American Mathematical Society
• 2020
We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers \$k\$, the largest possible diameter of a
Euler Polytopes and Convex Matroid Optimization
• Mathematics
• 2015
Del Pia and Michini recently improved the upper bound of kd due to Kleinschmidt and Onn for the largest possible diameter of the convex hull of a set of points in dimension d whose coordinates are

References

SHOWING 1-10 OF 25 REFERENCES
A counterexample to the Hirsch conjecture
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
• Mathematics
• 2010
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
• Mathematics
• 1992
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
Linear programming, the simplex algorithm and simple polytopes
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
• Mathematics
Math. Oper. Res.
• 1987
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
• Mathematics
• 1974
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