108 research outputs found
Complexity of some polyhedral enumeration problems
In this thesis we consider the problem of converting the halfspace representation of a polytope to its vertex representation - the Vertex Enumeration problem - and various other basic and closely related computational problems about polytopes. The problem of converting the vertex representation to halfspace representation - the Convex Hull problem - is equivalent to vertex enumeration. In chapter 3 we prove that enumerating the vertices of an unbounded H-polyhedron P is NP-hard even if P has only 0=1 vertices. This strengthens a previous result of Khachiyan et. al. [KBB+06]. In chapters 4 to 6 we prove that many other operations on polytopes like computing the Minkowski sum, intersection, projection, etc. are NP-hard for many representations and equivalent to vertex enumeration in many others. In chapter 7 we prove various hardness results about a cone covering problem where one wants to check whether a given set of polyhedral cones cover another given set. We prove that in general this is an NP-complete problem and includes important problems like vertex enumeration and hypergraph transversal as special cases. Finally, in chapter 8 we relate the complexity of vertex enumeration to graph isomorphism by proving that a certain graph isomorphism hard problem is graph isomorphism easy if and only if vertex enumeration is graph isomorphism easy. We also answer a question of Kaibel and Schwartz about the complexity of checking self-duality of a polytope.In dieser Arbeit betrachten wir das Problem, die Halbraumdarstellung eines Polytops in seine Eckendarstellung umzuwandeln, - das sogenannte Problem der EckenaufzĂ€hlung - sowie viele andere grundlegende und eng verwandte Berechnungsprobleme fĂŒr Polytope. Das Problem, die Eckendarstellung in die Halbraumdarstellung umzuwandeln - das sogenannte Konvexe-HĂŒllen Problem - ist Ă€quivalent zum Problem der EckenaufzĂ€hlung. In Kapitel 3 zeigen wir, dass EckenaufzĂ€hlung fĂŒr ein unbeschrĂ€nktes H-Polyeder P selbst dann NP-schwer ist, wenn P nur 0=1-Ecken hat. Das verbessert ein Ergebnis von Khachiyan et. al. [KBB+06]. In den Kapiteln 4 bis 6 zeigen wir, dass viele andere Operationen auf Polytopen, wie Berechnung von Minkowski-Summe, Durchschnitt, Projektion usw., fĂŒr viele Darstellungen NP-schwer sind und fĂŒr viele weitere Ă€quivalent zu EckenaufzĂ€hlung sind. In Kapitel 7 beweisen wir HĂ€rteresultate ĂŒber ein KegelĂŒberdeckungsproblem, das danach fragt, ob eine gegebene Menge polyedrischer Kegel eine andere gegebene Menge ĂŒberdeckt. Wir zeigen, dass dies im Allgemeinen ein NP-vollstĂ€ndiges Problem ist und wichtige Probleme wie EckenaufzĂ€hlung und Hypergraphentraversierung als SpezialfĂ€lle umfasst. SchlieĂlich stellen wir in Kapitel 8 einen Zusammenhang zwischen EckenaufzĂ€hlung und Graphisomorphie her, indem wir beweisen, dass ein bestimmtes Graphisomorphie-schweres Problem genau dann Graphisomorphie-leicht ist, wenn EckenaufzĂ€hlung Graphisomorphie-leicht ist. AuĂerdem beantworten wir eine Frage von Kaibel und Schwartz ĂŒber das Testen der Selbst-DualitĂ€t von Polytopen
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
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
The Hopf algebra of diagonal rectangulations
We define and study a combinatorial Hopf algebra dRec with basis elements
indexed by diagonal rectangulations of a square. This Hopf algebra provides an
intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter
permutations, which previously had only been described extrinsically as a sub
Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We
describe the natural lattice structure on diagonal rectangulations, analogous
to the Tamari lattice on triangulations, and observe that diagonal
rectangulations index the vertices of a polytope analogous to the
associahedron. We give an explicit bijection between twisted Baxter
permutations and the better-known Baxter permutations, and describe the
resulting Hopf algebra structure on Baxter permutations.Comment: Very minor changes from version 1, in response to comments by
referees. This is the final version, to appear in JCTA. 43 pages, 17 figure
Almost Symmetries and the Unit Commitment Problem
This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation
On 4-Dimensional Point Groups and on Realization Spaces of Polytopes
This dissertation consists of two parts. We highlight the main results from each part.
Part I. 4-Dimensional Point Groups. (based on a joint work with GĂŒnter Rote.)
We propose the following classification for the finite groups of orthogonal transformations in 4-space, the so-called 4-dimensional point groups.
Theorem A. The 4-dimensional point groups can be classified into
* 25 polyhedral groups (Table 5.1),
* 21 axial groups (7 pyramidal groups, 7 prismatic groups, and 7 hybrid groups, Table 6.3),
* 22 one-parameter families of tubical groups (11 left tubical groups and 11 right tubical groups, Table 3.1), and
* 25 infinite families of toroidal groups (2 three-parameter families, 19 two-parameter families, and 4 one-parameter families, Table 4.3.)
In contrast to earlier classifications of these groups (notably by Du Val in 1962 and by Conway and Smith in 2003), see Section 1.7), we emphasize a geometric viewpoint, trying to visualize and understand actions of these groups. Besides, we correct some omissions, duplications, and mistakes in these classifications.
The 25 polyhedral groups (Chapter 5) are related to the regular polytopes. The symmetries of the regular polytopes are well understood, because they are generated by reflections, and the classification of such groups as Coxeter groups is classic. We will deal with these groups only briefly, dwelling a little on just a few groups that come in enantiomorphic pairs (i.e., groups that are not equal to their own mirror.)
The 21 axial groups (Chapter 6) are those that keep one axis fixed. Thus, they essentially operate in the three dimensions perpendicular to this axis (possibly combined with a flip of the axis), and they are easy to handle, based on the well-known classification of the three-dimensional point groups.
The tubical groups (Chapter 3) are characterized as those that have (exactly) one Hopf bundle invariant. They come in left and right versions (which are mirrors of each other) depending on the Hopf bundle they keep invariant. They are so named because they arise with a decomposition of the 3-sphere into tube-like structures (discrete Hopf fibrations).
The toroidal groups (Chapter 4) are characterized as having an invariant torus. This class of groups is where our main contribution in terms of the completeness of the classification lies. We propose a new, geometric, classification of these groups. Essentially, it boils down to classifying the isometry groups of the two-dimensional square flat torus.
We emphasize that, regarding the completeness of the classification, in particular concerning the polyhedral and tubical groups, we rely on the classic approach (see Section 1.6). Only for the toroidal and axial groups, we supplant the classic approach by our geometric approach.
We give a self-contained presentation of Hopf fibrations (Chapter 2). In many places in the literature, one particular Hopf map is introduced as âthe Hopf mapâ, either in terms of four real coordinates or two complex coordinates, leading to âthe Hopf fibrationâ. In some sense, this is justified, as all Hopf bundles are (mirror-)congruent. However, for our characterization, we require the full generality of Hopf bundles. As a tool for working with Hopf fibrations, we introduce a parameterization for great circles in S^3 , which might be useful elsewhere.
Our main tool to understand tubical groups are polar orbit polytopes. (Chapter 1). In particular, we study the symmetries of a cell of the polar orbit polytope for different starting points.
Part II. Realization Spaces of Polytopes (based on a joint work with Rainer Sinn and GĂŒnter M. Ziegler.)
Robertson in 1988 suggested a model for the realization space of a d-dimensional polytope P, and an approach via the implicit function theorem to prove that the realization space is a smooth manifold of dimension NG(P) := d(f_0 + f_{dâ1} ) - f{0,d-1} . We call NG(P) the natural guess for (the dimension of the realization space of) P.
We build on Robertson's model and approach to study the realization spaces of higher-dimensional polytopes. We conclude combinatorial criteria (Sections 9.3.3 and 9.4.1) to decide if the realization space of the polytope in consideration is a smooth manifold of dimension given by the natural guess. As another application, we study the realization spaces of the second hypersimplices (Section 9.4.2).
We apply these criteria on 4-polytopes with small number of vertices, and along the way, we analyze examples where Robertsonâs approach fails, identifying the three smallest examples of 4-polytopes, for which the realization space is still a smooth manifold, but its dimension is not given by the natural guess (Section 9.5).
Finally, we investigate the realization space of the 24-cell (Section 9.5.2). We construct families of realizations of the 24-cell, and using them we show that the realization space of the 24-cell has points where it is not a smooth manifold. This provides the first known example of a polytope whose realization space is not a smooth manifold. We conclude by showing that the dimension of the realization space of the 24-cell is at least 48 and at most 52.Diese Dissertation befasst sich mit zwei verschiedenen Themen, von denen jedes seinen
eigenen Teil hat.
Der erste Teil befasst sich mit 4-dimensionalen Punktgruppen. Wir schlagen eine neue Klassifizierung fĂŒr diese Gruppen vor (siehe Theorem A), die im Gegensatz zu frĂŒheren Klassifizierungen eine geometrische Sichtweise betont und versucht, die Aktionen dieser Gruppen zu visualisieren und zu verstehen.
Im Folgenden werden diese Gruppen kurz beschrieben. Die polyedrischen Gruppen (Kapitel 5) sind mit den regelmĂ€Ăigen Polytopen verwandt. Die axialen Gruppen (Kapitel 6) sind diejenigen, die eine Achse festhalten. Die schlauchartigen Gruppen (Kapitel 3) werden als solche charakterisiert, die genau eine invariantes Hopf-BĂŒndel haben. Sie entstehen bei einer Zerlegung der 3-SphĂ€re in schlauchartige Strukturen (diskrete Hopf-Faserungen). Die toroidalen Gruppen (Kapitel 4) sind dadurch gekennzeichnet, dass sie einen invarianten Torus haben. Wir schlagen eine neue, geometrische Klassifizierung dieser Gruppen vor. Im Wesentlichen lĂ€uft sie darauf hinaus, die Isometriegruppen des zweidimensionalen quadratischen flachen Torus zu klassifizieren.
Nebenbei geben wir eine in sich geschlossene Darstellung der Hopf-Faserungen (Kapitel 2). Als Hilfsmittel fĂŒr die Arbeit mit ihnen fĂŒhren wir eine Parametrisierung fĂŒr GroĂkreise in S 3 ein, die an anderer Stelle nĂŒtzlich sein könnte.
Der zweite Teil befasst sich mit RealisierungsrÀumen von Polytopen. Wir bauen auf Robertsons Modell und Ansatz auf, um die RealisierungsrÀume von Polytopen zu untersuchen.
Wir stellen kombinatorische Kriterien auf (Abschnitte 9.3.3 und 9.4.1), um zu entscheiden, ob der Realisierungsraum des betrachteten Polytops eine glatte Mannigfaltigkeit der durch die ânatĂŒrliche Vermutungâ gegebenen Dimension ist. Als weitere Anwendung, untersuchen wir die RealisierungsrĂ€ume der zweiten Hypersimplices (Abschnitt 9.4.2).
Nebenbei identifizieren wir die kleinsten Beispiele von 4-Polytopen, fĂŒr die dieser Ansatz versagt (Abschnitt 9.5).
SchlieĂlich untersuchen wir den Realisierungsraum der 24-Zelle (Abschnitt 9.5.2). Wir zeigen, dass es Punkte gibt, an denen sie keine glatte Mannigfaltigkeit ist. Zuletzt zeigen wir, dass seine Dimension mindestens 48 und höchstens 52 betrĂ€gt
Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method
This dissertation investigates the geometric combinatorics of convex
polytopes and connections to the behavior of the simplex method for linear
programming. We focus our attention on transportation polytopes, which are sets
of all tables of non-negative real numbers satisfying certain summation
conditions. Transportation problems are, in many ways, the simplest kind of
linear programs and thus have a rich combinatorial structure. First, we give
new results on the diameters of certain classes of transportation polytopes and
their relation to the Hirsch Conjecture, which asserts that the diameter of
every -dimensional convex polytope with facets is bounded above by
. In particular, we prove a new quadratic upper bound on the diameter of
-way axial transportation polytopes defined by -marginals. We also show
that the Hirsch Conjecture holds for classical transportation
polytopes, but that there are infinitely-many Hirsch-sharp classical
transportation polytopes. Second, we present new results on subpolytopes of
transportation polytopes. We investigate, for example, a non-regular
triangulation of a subpolytope of the fourth Birkhoff polytope . This
implies the existence of non-regular triangulations of all Birkhoff polytopes
for . We also study certain classes of network flow polytopes
and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California,
Davis. 183 pages, 49 figure
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