707 research outputs found
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes
We develop an incremental algorithm to compute the Newton polytope
of the resultant, aka resultant polytope, or its
projection along a given direction.
The resultant is fundamental in algebraic elimination and
in implicitization of parametric hypersurfaces.
Our algorithm exactly computes vertex- and halfspace-representations
of the desired polytope using an oracle producing resultant vertices in a
given direction.
It is output-sensitive as it uses one oracle call per vertex.
We overcome the bottleneck of determinantal predicates
by hashing, thus accelerating execution from to times.
We implement our algorithm using the experimental CGAL package {\tt
triangulation}.
A variant of the algorithm computes successively tighter inner and outer
approximations: when these polytopes have, respectively,
90\% and 105\% of the true volume, runtime is reduced up to times.
Our method computes instances of -, - or -dimensional polytopes
with K, K or vertices, resp., within hr.
Compared to tropical geometry software, ours is faster up to
dimension or , and competitive in higher dimensions
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
Maximal admissible faces and asymptotic bounds for the normal surface solution space
The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint. The main results of this
paper are significant improvements upon the best known asymptotic bounds on the
number of admissible vertices, using polytopes in both the standard normal
surface coordinate system and the streamlined quadrilateral coordinate system.
To achieve these results we examine the layout of admissible points within
these polytopes. We show that these points correspond to well-behaved
substructures of the face lattice, and we study properties of the corresponding
"admissible faces". Key lemmata include upper bounds on the number of maximal
admissible faces of each dimension, and a bijection between the maximal
admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in
Journal of Combinatorial Theory A
Graphs of Transportation Polytopes
This paper discusses properties of the graphs of 2-way and 3-way
transportation polytopes, in particular, their possible numbers of vertices and
their diameters. Our main results include a quadratic bound on the diameter of
axial 3-way transportation polytopes and a catalogue of non-degenerate
transportation polytopes of small sizes. The catalogue disproves five
conjectures about these polyhedra stated in the monograph by Yemelichev et al.
(1984). It also allowed us to discover some new results. For example, we prove
that the number of vertices of an transportation polytope is a
multiple of the greatest common divisor of and .Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of
several lemmas and the upper bound in Theorem 1.1 is improved by a factor of
tw
Amplituhedron meets Jeffrey-Kirwan Residue
The tree amplituhedra A^(m)_n,k are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed for m=4 as a geometric construction encoding tree-level scattering amplitudes in planar N=4 super Yang-Mills theory, they are mathematically interesting for any m. In this paper we strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. We focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional conjugates. We show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron volume functions in all these cases. Notably, this also naturally exposes the rich combinatorial and geometric structures of amplituhedra, such as their regular triangulations.Peer reviewedFinal Accepted Versio
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