548 research outputs found
One-Point Suspensions and Wreath Products of Polytopes and Spheres
It is known that the suspension of a simplicial complex can be realized with
only one additional point. Suitable iterations of this construction generate
highly symmetric simplicial complexes with various interesting combinatorial
and topological properties. In particular, infinitely many non-PL spheres as
well as contractible simplicial complexes with a vertex-transitive group of
automorphisms can be obtained in this way.Comment: 17 pages, 8 figure
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
On the sum of the Voronoi polytope of a lattice with a zonotope
A parallelotope is a polytope that admits a facet-to-facet tiling of
space by translation copies of along a lattice. The Voronoi cell
of a lattice is an example of a parallelotope. A parallelotope can be
uniquely decomposed as the Minkowski sum of a zone closed parallelotope and
a zonotope , where is the set of vectors used to generate the
zonotope. In this paper we consider the related question: When is the Minkowski
sum of a general parallelotope and a zonotope a parallelotope? We give
two necessary conditions and show that the vectors have to be free. Given a
set of free vectors, we give several methods for checking if is
a parallelotope. Using this we classify such zonotopes for some highly
symmetric lattices.
In the case of the root lattice , it is possible to give a more
geometric description of the admissible sets of vectors . We found that the
set of admissible vectors, called free vectors, is described by the well-known
configuration of lines in a cubic. Based on a detailed study of the
geometry of , we give a simple characterization of the
configurations of vectors such that is a
parallelotope. The enumeration yields maximal families of vectors, which
are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table
Shadows of Newton Polytopes
We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity
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