Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d = 6. This implies that for all pairs (d, n) with n−d ≤ 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n − d ≤ 6. We show this result by showing this bound for a more general structure – so-called matroid polytopes – by reduction to a small number of satisfiability problems. This article deals with the determination of the maximal diameter ∆(d, n) of a d-dimensional polytope with n facets. The Hirsch Conjecture states that ∆(d, n) ≤ n−d. The special case that ∆(d, 2d) ≤ d is known as the d-step conjecture. Showing the d-step conjecture for a fixed d implies (by an argument of Klee and Walkup [9]) that the Hirsch conjecture holds for all pairs (n, d) with n−d = d. So establishing the d-step conjecture for all d implies the general Hirsch conjecture. So far, the d-step conjecture has been shown for all d ≤ 5 by Klee and Walkup. We show that the d-step conjecture is true in dimension 6 as well. We derive this result by considering a more general class of objects, namely matroid polytopes, i.e. oriented matroids, which, if realizable, correspond to convex polytopes. We show that no 6-dimensional matroid polytope with 12 vertices and a facet path of length 7 exists. Then ∆(6, 12) = 6 follows by considering polarity and the already known bounds. To show that ∆(6, 12) ≤ 6 we first give combinatorial conditions for matroid polytopes that violate this bound. This achieved through the study of path complexes (see Bremner et al. [2], cf. Section 1). We then show that these conditions can not be satisfied by an oriented matroid. To show this we use a satisfiability solver to produce the desired contradiction (see Section 2). We will use the same method to show that ∆(4, 11) = 6, which settles another special case of the Hirsch conjecture. The latter result allows us to also improve the upper bound on ∆(5, 12) from 9 to 8. For small parameters there are already known general bounds that allow us to compute or at least bound the diameter of polytopes. We summarize them in the following Lemma. For an overview about the known bounds we refer to the respective chapters in the books by Grünbaum [5] and Ziegler [13] and the survey of Klee and Kleinschmidt [8]. Lemma 1 (Klee [7], Klee and Walkup [9], Holt [6]). The following relations hold for the maximal diameter ∆(d, n) of a d-polytope with n facets: