10 research outputs found
Perturbation of transportation polytopes
We describe a perturbation method that can be used to reduce the problem of
finding the multivariate generating function (MGF) of a non-simple polytope to
computing the MGF of simple polytopes. We then construct a perturbation that
works for any transportation polytope. We apply this perturbation to the family
of central transportation polytopes of order kn x n, and obtain formulas for
the MGFs of the feasible cone of each vertex of the polytope and the MGF of the
polytope. The formulas we obtain are enumerated by combinatorial objects. A
special case of the formulas recovers the results on Birkhoff polytopes given
by the author and De Loera and Yoshida. We also recover the formula for the
number of maximum vertices of transportation polytopes of order kn x n.Comment: 25 pages, 3 figures. To appear in Journal of Combinatorial Theory
Ser.
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Ehrhart Positivity for Generalized Permutohedra
International audienceThere are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne.Il existe peu de résultats sur les coefficients des polynômes d’Ehrhart. On présente une conjecture concernant leur positivité pour une certaine famille de polytopes connus sous le nom de permutoèdre généralisé. On a vérifié la conjecture pour les petites dimensions en combinant des méthodes de perturbation avec une nouvelle valuation sur l’algèbre des cônes polyédraux rationnels pointés, construite par Berline et Vergne
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Ehrhart Positivity for Generalized Permutohedra
There are few general results about the coefficients of Ehrhart polynomials. We present a conjecture about their positivity for a certain family of polytopes known as generalized permutohedra. We have verified the conjecture for small dimensions combining perturbation methods with a new valuation on the algebra of rational pointed polyhedral cones constructed by Berline and Vergne
Cliques and independent sets of the Birkhoff polytope graph
The Birkhoff polytope graph has a vertex set equal to the elements of the
symmetric group of degree , and two elements are adjacent if one element
equals the product of the other element with a cycle. Maximal and maximum
cliques (sets of pairwise adjacent elements) and independent sets (sets of
pairwise nonadjacent elements) of the Birkhoff polytope graph are studied.
Bounds are obtained for different sizes of such sets
Perturbation of transportation polytopes
We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order , we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order
Recommended from our members
Perturbation of transportation polytopes
We describe a perturbation method that can be used to reduce the problem of finding
the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of
simple polytopes. We then construct a perturbation that works for any transportation
polytope. We apply this perturbation to the family of central transportation polytopes of
order kn x n, and obtain formulas for the MGFs of the feasible cone of each vertex of the
polytope and the MGF of the polytope. The formulas we obtain are enumerated by
combinatorial objects. A special case of the formulas recovers the results on Birkhoff
polytopes given by the author and De Loera and Yoshida. We also recover the formula for the
number of maximum vertices of transportation polytopes of order kn x n