Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

Details Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

2008-09-05

arxiv.org/abs/0809.0915v3

Mathematics

Combinatorics

9 pages; update shortcut constraint discussion

Scientific paper

10.1080/10586458.2011.564965

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.

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