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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 \leq 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 \leq 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.Comment: 9 pages; update shortcut constraint discussio