Improving bounds on the diameter of a polyhedron in high dimensions

Abstract

In 1992, Kalai and Kleitman proved that the diameter of a $d$-dimensional polyhedron with $n$ facets is at most $n^{2+\log_2 d}$. In 2014, Todd improved the Kalai-Kleitman bound to $(n-d)^{\log_2 d}$. We improve the Todd bound to $(n-d)^{-1+\log_2 d}$ for $n \ge d \ge 7$, $(n-d)^{-2+\log_2 d}$ for $n \ge d \ge 37$, and $(n-d)^{-3+\log_2 d+O\left(1/d\right)}$ for $n \ge d \ge 1$

The diameter of lattice zonotopes

“…Another quantity that has attracted attention, due to its connection with the complexity of the simplex algorithm [15,23,26,27], is the largest diameter δ(d, k) a lattice polytope contained in [0, k] d can have [11,12,13,17,20]. Here, by the diameter of a polytope, we mean the diameter of the graph made of its vertices and edges.…”

Distance between vertices of lattice polytopes

“…Larman [15] gave an upper bound on this quantity that is linear as a function of n, but exponential as a function of d, which was subsequently refined by Barnette [3] and generalized by Eisenbrand, Hähnle, Razborov, and Rothvoß [9] and Labbé, Manneville, and Santos [14]. Kalai and Kleitman [11] found an upper bound that is quasi-polynomial as a function of d and n, which was subsequently refined by Todd [20] and Sukegawa [18]. Lower bounds have also been obtained by Klee and Walkup [12] and by Santos [17], disproving the Hirsch conjecture for unbounded polyhedra and for polytopes, respectively.…”

On Circuit Diameter Bounds via Circuit Imbalances

“…The first quasipolynomial bound was given by Kalai and Kleitman [Kal92,KK92], see [Suk17] for the best current bound and an overview of the literature. Dyer and Frieze [DF94] showed the polynomial Hirsch conjecture for totally unimodular (TU) matrices.…”