2 research outputs found
Bounds on the number of 2-level polytopes, cones and configurations
We prove an upper bound of the form on the
number of affine (resp. linear) equivalence classes of, by increasing order of
generality, 2-level d-polytopes, d-cones and d-configurations. This in
particular answers positively a conjecture of Bohn et al. on 2-level polytopes.
We obtain our upper bound by relating affine (resp. linear) equivalence classes
of 2-level d-polytopes, d-cones and d-configurations to faces of the
correlation cone. We complement this with a lower bound, by
estimating the number of nonequivalent stable set polytopes of bipartite
graphs.Comment: 10 page
Small Shadows of Lattice Polytopes
The diameter of the graph of a -dimensional lattice polytope is known to be at most due to work by Kleinschmidt and Onn.
However, it is an open question whether the monotone diameter, the shortest
guaranteed length of a monotone path, of a -dimensional lattice polytope is bounded
by a polynomial in and . This question is of particular interest in
linear optimization, since paths traced by the Simplex method must be monotone.
We introduce partial results in this direction including a monotone diameter
bound of for , a monotone diameter bound of for
-dimensional -level polytopes, a pivot rule such that the Simplex
method is guaranteed to take at most non-degenerate steps
to solve a LP on , and a bound of for lengths of paths from certain
fixed starting points. Finally, we present a constructive approach to a
diameter bound of and describe how to translate this final bound into
an algorithm that solves a linear program by tracing such a path.Comment: 11 page