2,271 research outputs found
Polymake and Lattice Polytopes
The polymake software system deals with convex polytopes and related objects
from geometric combinatorics. This note reports on a new implementation of a
subclass for lattice polytopes. The features displayed are enabled by recent
changes to the polymake core, which will be discussed briefly.Comment: 12 pages, 1 figur
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Faces of highest weight modules and the universal Weyl polyhedron
Let be a highest weight module over a Kac-Moody algebra ,
and let conv denote the convex hull of its weights. We determine the
combinatorial isomorphism type of conv , i.e. we completely classify the
faces and their inclusions. In the special case where is
semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN
2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans.
Amer. Math. Soc. 2017] for most modules. The determination of faces of
finite-dimensional modules up to the Weyl group action and some of their
inclusions also appears in previous work of Satake [Ann. of Math. 1960],
Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and
Casselman [Austral. Math. Soc. 1997].
For any subset of the simple roots, we introduce a remarkable convex cone
which we call the universal Weyl polyhedron, which controls the convex hulls of
all modules parabolically induced from the corresponding Levi factor. Namely,
the combinatorial isomorphism type of the cone stores the classification of
faces for all such highest weight modules, as well as how faces degenerate as
the highest weight gets increasingly singular. To our knowledge, this cone is
new in finite and infinite type.
We further answer a question of Michel Brion, by showing that the
localization of conv along a face is always the convex hull of the weights
of a parabolically induced module. Finally, as we determine the inclusion
relations between faces representation-theoretically from the set of weights,
without recourse to convexity, we answer a similar question for highest weight
modules over symmetrizable quantum groups.Comment: Final version, to appear in Advances in Mathematics (42 pages, with
similar margins; essentially no change in content from v2). We recall
preliminaries and results from the companion paper arXiv:1606.0964
Basic Polyhedral Theory
This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of
Operations Research and Management Science) describing parts of the theory of
convex polyhedra that are particularly important for optimization. The topics
include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem,
faces of polyhedra, projections of polyhedra, integral polyhedra, total dual
integrality, and total unimodularity.Comment: 14 page
A-Tint: A polymake extension for algorithmic tropical intersection theory
In this paper we study algorithmic aspects of tropical intersection theory.
We analyse how divisors and intersection products on tropical cycles can
actually be computed using polyhedral geometry. The main focus of this paper is
the study of moduli spaces, where the underlying combinatorics of the varieties
involved allow a much more efficient way of computing certain tropical cycles.
The algorithms discussed here have been implemented in an extension for
polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be
published in European Journal of Combinatoric
Approximation of corner polyhedra with families of intersection cuts
We study the problem of approximating the corner polyhedron using
intersection cuts derived from families of lattice-free sets in .
In particular, we look at the problem of characterizing families that
approximate the corner polyhedron up to a constant factor, which depends only
on and not the data or dimension of the corner polyhedron. The literature
already contains several results in this direction. In this paper, we use the
maximum number of facets of lattice-free sets in a family as a measure of its
complexity and precisely characterize the level of complexity of a family
required for constant factor approximations. As one of the main results, we
show that, for each natural number , a corner polyhedron with basic
integer variables and an arbitrary number of continuous non-basic variables is
approximated up to a constant factor by intersection cuts from lattice-free
sets with at most facets if and that no such approximation is
possible if . When the approximation factor is allowed to
depend on the denominator of the fractional vertex of the linear relaxation of
the corner polyhedron, we show that the threshold is versus .
The tools introduced for proving such results are of independent interest for
studying intersection cuts
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