631 research outputs found
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Vertex-Facet Incidences of Unbounded Polyhedra
How much of the combinatorial structure of a pointed polyhedron is contained
in its vertex-facet incidences? Not too much, in general, as we demonstrate by
examples. However, one can tell from the incidence data whether the polyhedron
is bounded. In the case of a polyhedron that is simple and "simplicial," i.e.,
a d-dimensional polyhedron that has d facets through each vertex and d vertices
on each facet, we derive from the structure of the vertex-facet incidence
matrix that the polyhedron is necessarily bounded. In particular, this yields a
characterization of those polyhedra that have circulants as vertex-facet
incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure
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