A refinement of Todd’s bound for the diameter of a polyhedron

Improving bounds on the diameter of a polyhedron in high dimensions

“…Refining Kalai and Kleitman's approach, in [29], Todd showed in 2014 that ∆(d, n) ≤ (n − d) log 2 d for n ≥ d ≥ 1. The Todd bound is tight for d ≤ 2 and coincides with the true value ∆(d, d), i.e., 0, when n = d. Sukegawa and Kitahara [28] slightly improved the Todd bound to (n − d) log 2 (d−1) for n ≥ d ≥ 3. We note that their bound is no longer valid for d ≤ 2, however, it coincides with the Hirsch bound of n − d, and is tight for d = 3.…”

“…Remark 6. Since our proof method is based on only the Kalai-Kleitman inequality and the generalized Larman bound, one can easily apply it to a more generalized setting where we have the similar results; see, e.g., [10] who proved an improved upper bound on the diameter of normal simplicial complexes by extending the proof of [28], a special case of this study.…”

“…As in [12,28,29], each inequality stated in Theorem 1 will be proved via an induction on d based on the Kalai-Kleitman inequality. In contrast to [12,28,29], we introduce a way of strengthening Todd's analysis for the inductive step in high dimensions. In this approach, on the other hand, we need to check a large number of pairs (d, n) for the base case.…”

Tail diameter upper bounds for polytopes and polyhedra

“…The best known upper bounds on the diameters of convex polyhedra are the quasipolynomial bounds given by Kalai and Kleitman (see [19]), with recent improvements by Todd (see [40]) and Sukegawa and Kitahara (see [39]). For polytopes, there are several classical upper bounds on the diameters of polytopes which are linear in the number n of facets of P given by Barnette (see [6] and [7]) and Larman (see [29]).…”

A simple proof of tail--polynomial bounds on the diameter of polyhedra

“…The former is due to Kalai and Kleitman [2], which has been used for proving bounds on ∆(d, n) for years, see, e.g., [2,12,10,11,1]. The latter is by Klee and Walkup [6].…”