1,013 research outputs found
Fundamental polytopes of metric trees via parallel connections of matroids
We tackle the problem of a combinatorial classification of finite metric
spaces via their fundamental polytopes, as suggested by Vershik in 2010. In
this paper we consider a hyperplane arrangement associated to every split
pseudometric and, for tree-like metrics, we study the combinatorics of its
underlying matroid. We give explicit formulas for the face numbers of
fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we
characterize the metric trees for which the fundamental polytope is simplicial.Comment: 20 pages, 2 Figures, 1 Table. Exposition improved, references and new
results (last section) adde
Affine Symmetries of Orbit Polytopes
An orbit polytope is the convex hull of an orbit under a finite group . We develop a general theory of possible
affine symmetry groups of orbit polytopes. For every group, we define an open
and dense set of generic points such that the orbit polytopes of generic points
have conjugated affine symmetry groups. We prove that the symmetry group of a
generic orbit polytope is again if is itself the affine symmetry group
of some orbit polytope, or if is absolutely irreducible. On the other hand,
we describe some general cases where the affine symmetry group grows.
We apply our theory to representation polytopes (the convex hull of a finite
matrix group) and show that their affine symmetries can be computed effectively
from a certain character. We use this to construct counterexamples to a
conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math.
222 (2009), 431--452, Conjecture~5.4].Comment: v2: Referee comments implemented, last section updated. Numbering of
results changed only in Sections 9 and 10. v3: Some typos corrected. Final
version as published. 36 pages, 5 figures (TikZ
Ehrhart clutters: Regularity and Max-Flow Min-Cut
If C is a clutter with n vertices and q edges whose clutter matrix has column
vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a
Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to
show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then
C is an Ehrhart clutter and in this case we provide sharp bounds on the
Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols
conjecture on packing problems, we conjecture that if C is both ideal and the
clique clutter of a perfect graph, then C has the MFMC property. We prove this
conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel
graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof
of our conjecture when C is a uniform clique clutter of a perfect graph. We
close with a generalization of Ehrhart clutters as it relates to total dual
integrality.Comment: Electronic Journal of Combinatorics, to appea
Positive configuration space
We define and study the totally nonnegative part of the Chow quotient of the
Grassmannian, or more simply the nonnegative configuration space. This space
has a natural stratification by positive Chow cells, and we show that
nonnegative configuration space is homeomorphic to a polytope as a stratified
space. We establish bijections between positive Chow cells and the following
sets: (a) regular subdivisions of the hypersimplex into positroid polytopes,
(b) the set of cones in the positive tropical Grassmannian, and (c) the set of
cones in the positive Dressian. Our work is motivated by connections to super
Yang-Mills scattering amplitudes, which will be discussed in a sequel.Comment: 46 pages; citations adde
Tropical cycles and Chow polytopes
The Chow polytope of an algebraic cycle in a torus depends only on its
tropicalisation. Generalising this, we associate a Chow polytope to any
abstract tropical variety in a tropicalised toric variety. Several significant
polyhedra associated to tropical varieties are special cases of our Chow
polytope. The Chow polytope of a tropical variety is given by a simple
combinatorial construction: its normal subdivision is the Minkowski sum of
and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties;
several improvements suggested by Eric Katz; some rearrangemen
Generalized Permutohedra from Probabilistic Graphical Models
A graphical model encodes conditional independence relations via the Markov
properties. For an undirected graph these conditional independence relations
can be represented by a simple polytope known as the graph associahedron, which
can be constructed as a Minkowski sum of standard simplices. There is an
analogous polytope for conditional independence relations coming from a regular
Gaussian model, and it can be defined using multiinformation or relative
entropy. For directed acyclic graphical models and also for mixed graphical
models containing undirected, directed and bidirected edges, we give a
construction of this polytope, up to equivalence of normal fans, as a Minkowski
sum of matroid polytopes. Finally, we apply this geometric insight to construct
a new ordering-based search algorithm for causal inference via directed acyclic
graphical models.Comment: Appendix B is expanded. Final version to appear in SIAM J. Discrete
Mat
Average case polyhedral complexity of the maximum stable set problem
We study the minimum number of constraints needed to formulate random
instances of the maximum stable set problem via linear programs (LPs), in two
distinct models. In the uniform model, the constraints of the LP are not
allowed to depend on the input graph, which should be encoded solely in the
objective function. There we prove a lower bound with
probability at least for every LP that is exact for a randomly
selected set of instances; each graph on at most n vertices being selected
independently with probability . In the
non-uniform model, the constraints of the LP may depend on the input graph, but
we allow weights on the vertices. The input graph is sampled according to the
G(n, p) model. There we obtain upper and lower bounds holding with high
probability for various ranges of p. We obtain a super-polynomial lower bound
all the way from to . Our upper bound is close to this as there is only an essentially quadratic
gap in the exponent, which currently also exists in the worst-case model.
Finally, we state a conjecture that would close this gap, both in the
average-case and worst-case models
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