44 research outputs found
Tight Lower Bounds on the Sizes of Symmetric Extensions of Permutahedra and Similar Results
It is well known that the permutahedron Pi_n has 2^n-2 facets. The Birkhoff
polytope provides a symmetric extended formulation of Pi_n of size Theta(n^2).
Recently, Goemans described a non-symmetric extended formulation of Pi_n of
size Theta(n log(n)). In this paper, we prove that Omega(n^2) is a lower bound
for the size of symmetric extended formulations of Pi_n.Comment: corrected an error in the linear description of the permutahedron in
introductio
Hidden Vertices in Extensions of Polytopes
Some widely known compact extended formulations have the property that each
vertex of the corresponding extension polytope is projected onto a vertex of
the target polytope. In this paper, we prove that for heptagons with vertices
in general position none of the minimum size extensions has this property.
Additionally, for any d >= 2 we construct a family of d-polytopes such that at
least 1/9 of all vertices of any of their minimum size extensions is not
projected onto vertices.Comment: 9 pages, to appear in: Operations Research Letter
Uncapacitated Flow-based Extended Formulations
An extended formulation of a polytope is a linear description of this
polytope using extra variables besides the variables in which the polytope is
defined. The interest of extended formulations is due to the fact that many
interesting polytopes have extended formulations with a lot fewer inequalities
than any linear description in the original space. This motivates the
development of methods for, on the one hand, constructing extended formulations
and, on the other hand, proving lower bounds on the sizes of extended
formulations.
Network flows are a central paradigm in discrete optimization, and are widely
used to design extended formulations. We prove exponential lower bounds on the
sizes of uncapacitated flow-based extended formulations of several polytopes,
such as the (bipartite and non-bipartite) perfect matching polytope and TSP
polytope. We also give new examples of flow-based extended formulations, e.g.,
for 0/1-polytopes defined from regular languages. Finally, we state a few open
problems
The Projected Faces Property and Polyhedral Relations
Margot (1994) in his doctoral dissertation studied extended formulations of
combinatorial polytopes that arise from "smaller" polytopes via some
composition rule. He introduced the "projected faces property" of a polytope
and showed that this property suffices to iteratively build extended
formulations of composed polytopes.
For the composed polytopes, we show that an extended formulation of the type
studied in this paper is always possible only if the smaller polytopes have the
projected faces property. Therefore, this produces a characterization of the
projected faces property.
Affinely generated polyhedral relations were introduced by Kaibel and
Pashkovich (2011) to construct extended formulations for the convex hull of the
images of a point under the action of some finite group of reflections. In this
paper we prove that the projected faces property and affinely generated
polyhedral relation are equivalent conditions