More bounds on the diameters of convex polytopes

Abstract

Let ∆(d, n) be the maximum possible edge diameter over all d-dimensional polytopes defined by n inequalities. The Hirsch conjecture, formulated in 1957, suggests that ∆(d, n) is no greater than n − d. No polynomial bound is currently known for ∆(d, n), the best one being quasipolynomial due to Kalai and Kleitman in 1992. Goodey showed in 1972 that ∆(4, 10) = 5 and ∆(5, 11) = 6, and more recently, Bremner and Schewe showed ∆(4, 11) = ∆(6, 12) = 6. In this follow-up, we show that ∆(4, 12) = 7 and present strong evidence that ∆(5, 12) = ∆(6, 13) = 7.

A Proof of the Strict Monotone 5-Step Conjecture

“…With the last two entries added, we have the circuit (0, +, −, −, +, 0, 0, 0, −, 0, +, ?) and cocircuit (+, 0, 0, 0, 0, +, +, +, 0, +, ?, +) This implies, by Theorem 2.1, the following conditions on χ : χ(2, 3, 4, 5, 9, 1) = χ (2,3,4,5,9,12), χ(2, 3, 4, 5, 9, 6) = χ (2,3,4,5,9,12), χ(2, 3, 4, 5, 9, 7) = χ (2,3,4,5,9,12), χ(2, 3, 4, 5, 9, 8) = χ(2, 3, 4, 5, 9, 12), χ(2, 3, 4, 5, 9, 10) = χ (2,3,4,5,9,12), from the cocircuit, and χ(2, 3, 4, 5, 9, 12) = −χ (11,3,4,5,9,12), χ(2, 3, 4, 5, 9, 12) = χ (2,11,4,5,9,12), χ(2, 3, 4, 5, 9, 12) = χ(2, 3, 11, 5, 9, 12), χ(2, 3, 4, 5, 9, 12) = −χ (2,3,4,11,9,12), χ(2, 3, 4, 5, 9, 12) = χ(2, 3, 4, 5, 11, 12), from the circuit. Each of these conditions can be enforced by two clauses with two literals each.…”

“…Such a counterexample would lead to a function χ satisfying 10 constraints each for these two vertices. The constraints χ((1, 2, 3, 4, 5), 11) = χ((1, 2, 3, 4, 5) k,12 , 11) for k = 1, 2, 3, 4, 5, together imply that [1,2,3,4,5] is a vertex of the oriented matroid program, where (1, 2, 3, 4, 5) k,12 is the ordered set (1, 2, 3, 4, 5) with entry k replaced by 12. Similarly, the constraints χ((6, 7, 8, 9, 10), 11) = χ((6, 7, 8, 9, 10) k,12 , 11) for k = 6, 7, 8, 9, 10 imply that [6,7,8,9,10] is a vertex.…”

“…This orientation satisfied some combinatorial conditions that admissible orientations must satisfy, such as having a unique source and sink on every face. Bremner and Schewe [2], [3] showed how to use satisfiability solvers to determine if there existed a counterexample to the 6-step conjecture ∆(6, 12) = 6. They showed that that not only did there not exist such a polytope, but there existed no matroid polytope that could serve as the dual of a combinatorial counterexample.…”

A Flag Vector of a 3‐sphere That Is Not the Flag Vector of a 4‐polytope

“…8]). However, as far as we know this has always been applied to simplicial polytopes or surfaces, and thus in a setting of uniform oriented matroids, with the notable exception of Bremner's software package mpc [12]; see Bokowski et al [9, §7]. (Indeed, David Bremner has confirmed the non‐existence result of this section using the nuoms function of his package.)…”

A counterexample to the Hirsch Conjecture

“…In a previous paper [31], Klee had shown the Hirsch Conjecture for d = 3. Together with the cases (n, d) ∈ {(11, 4), (12, 4)} [8], [9], these are all the parameters where the Hirsch Conjecture is known to hold.…”