DOCSLIB.ORG

Extended Formulations for Combinatorial Polytopes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Extended Formulations for Combinatorial Polytopes Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) von Kanstantsin Pashkovich geb. am 06.07.1985 in Bobruisk genehmigt durch die Fakultät für Mathematik der Otto-von-Guericke-Universität Magdeburg Gutachter: Prof. Dr. Volker Kaibel Prof. Dr. Michelangelo Conforti eingereicht am: 01.02.2012 Verteidigung am: 31.08.2012 1 ABSTRACT i Abstract Typically polytopes arising from real world problems have a lot of facets. In some cases even no linear descriptions for them are known. On the other hand many of these polytopes can be described much nicer and with less facets using extended formulations, i.e. as a projection of simpler higher dimensional polytopes. The presented work studies extended formulations for polytopes: the possibilities to construct extended formulations and limitations of them. In the first part, some known techniques for constructions of extended formulations are reviewed and the new framework of polyhedral relations (see Kaibel and Pashkovich [2011]) is presented. We in particular elaborate on the special case of reflection relations. Reflection relations provide extended formulations for several polytopes that can be con- structed by iteratively taking convex hulls of polytopes and their refelections at hyper- planes. Using this framework we are able to derive small extended formulations for the G-permutahedra of all finite reflection groups G. The second part deals with extended formulations which use special structures of graphs involved in combinatorial problems. Here we present some known extended for- mulations and apply a few changes to the extended formulation of Gerards for the perfect matching polytope in graphs with small genus in order to reduce its size. Furthermore a new compact proof of an extended formulation of Rivin for the subtour elimination poly- tope is provided. The third part (partly based on joint work with Volker Kaibel, Samuel Fiorini and Dirk Oliver Theis, see Fiorini, Kaibel, Pashkovich, and Theis [2011a]) involves general ques- tions on the extended formulations of polytopes. The primal interest here are lower bounds for extended formulations. We study different techniques to obtain these lower bounds, all of which could be derived from so called non-negative factorizations of the slack matrix of the initial polytope. The minimal such factorization provides the minimal number of in- equalities needed in an extended formulation. We compare different techniques, find their limitations and provide examples of the polytopes for which they give tight lower bounds on the complexity of extensions. The fourth part studies the impact of symmetry on the sizes of extended formulations. In joint work with Volker Kaibel and Dirk Oliver Theis we showed that for certain con- strained cardinality matching and cycle polytopes there exist no polynomial symmetric extended formulations, but there are polynomial non-symmetric ones (for further details see Kaibel, Pashkovich, and Theis [2010]). Beyond these results the thesis also contains a proof showing that the well known symmetric extended formulation for the permutahe- dron via the Birkhoff polytope is the best (up to a constant factor) one among symmetric extended formulations (see Pashkovich [2009]). ii Zusammenfassung Viele kombinatorische Polytope, die ihre Anwendung in praktischen Problemen fin- den, haben eine große Anzahl von Facetten. In manchen Beispielen ist nicht einmal ei- ne lineare Beschreibung dieser Polytope bekannt. Andererseits lassen viele Polytope eine kompakte und schönere Darstellung mit Hilfe von erweiterten Formulierungen (d.h. als Projektion einfacherer höher-dimensionaler Polytope) zu. Die vorgelegte Arbeit untersucht Erweiterungen von Polytopen: Möglichkeiten, eine kompakte Erweiterung zu finden, und Einschränkungen dieses Ansatzes. Im ersten Teil, werden einige bekannte Konstruktionen für Erweiterungen dargestellt und das neue Framework der Polyedrischen Relationen eingeführt, welches Teil einer ge- meinsamen Arbeit mit Volker Kaibel ist (siehe Kaibel and Pashkovich [2011]). Insbeson- dere arbeiten wir den Fall von Spiegelungsrelationen aus. Spiegelungsrelationen liefern erweiterte Formulierungen für Polytope, die durch iterierte Bildung konvexer Hüllen von Polytopen und ihrer Spiegelungen an Hyperebenen konstruiert werden können. Mit Hil- fe dieses Frameworks können wir kompakte Erweiterungen von G-Permutaedern für alle endliche Spiegelungsgruppen G konstruieren. Der zweite Teil beschäftigt sich mit Erweiterungen, welche spezielle Eigenschaften von den Graphen ausnutzen, die in kombinatorischen Problemen auftauchen. Hier präsen- tieren wir einige bekannte Erweiterungen, und nehmen kleine Änderungen in der erwei- terten Formulierung von Gerards für Perfekte Matching Polytope in Graphen mit kleinem Geschlecht vor, um die Grösse der Erweiterung zu reduzieren. Dazu präsentieren wir einen einfachen Beweis für die Erweiterung von Rivin des Subtour Elimination Polytops. Der dritte Teil untersucht prinzipielle Grenzen des Konzepts der erweiterten Formulie- rungen. Das Hauptziel des dritten Teils ist es, untere Schranken für die Größe erweiterter Formulierung herzustellen. Hier stellen wir einige gemeinsame Ergebnisse mit Volker Kai- bel, Samuel Fiorini und Dirk Oliver Theis dar (siehe Fiorini, Kaibel, Pashkovich, and Theis [2011a]). Wir vergleichen verschiedene Methoden, um untere Schranken zu bekommen, und schauen uns verschiedene Beispiele von Polytopen an (für manche Polytope sind die Schranken optimal). Im vierten Teil präsentieren wir eine weitere gemeinsame Arbeit mit Volker Kai- bel und Dirk Oliver Theis, die sich mit Symmetrien in Erweiterungen beschäftigt (se- he Kaibel, Pashkovich, and Theis [2010]). Hier haben wir Matching und Cycle Polytope gefunden, welche keine symmetrischen Erweiterungen haben, aber sich trotzdem mit Hilfe von Erweiterungen kompakt darstellen lassen. Über gemeinsame Arbeit hinaus, beweisen wir Ergebnisse bezüglich Erweiterungen von quadratischer Grösse, welche zum Beispiel zeigen, dass das Birkhoff Polytop eine asymptotisch minimale symmetrische Erweiterung des Permutaeders ist (sehe Pashkovich [2009]). Contents Abstract i Zusammenfassung ii Chapter 1. Introduction 1 Acknowledgments 2 1.1. Preliminaries 3 1.1.1. Polytopes 3 1.1.2. Extended Formulations, Extensions 4 1.1.3. Combinatorial Polytopes 5 1.1.3.1. Spanning Tree Polytope 5 1.1.3.2. Matching Polytope 5 1.1.3.3. Cycle Polytope 6 1.2. Extensions of Combinatorial Polytopes 6 Chapter 2. Balas Extensions, Flow Extensions and Polyhedral Relations 9 2.1. Balas Extensions 9 2.2. Dynamic Programming Extensions 11 2.3. Flow Extensions 11 2.4. Cardinality Indicating Polytope 12 2.5. Parity Polytope 14 2.6. Birkhoff Polytope and Perfect Matchings in Bipartite Graphs 15 2.7. Permutahedron 16 2.8. Edge Polytope 16 2.9. Cardinality Restricted Matching Polytopes 17 2.10. Cardinality Restricted Cycle Polytope 18 2.11. Polyhedral Relations 18 2.12. Sequential Polyhedral Relations 19 2.13. Affinely Generated Polyhedral Relations 20 2.14. Affine Generators and Domains from Polyhedral Relation 21 2.15. Polyhedral Relations from Affine Generators and Domains 22 2.16. Reflection Relations 23 2.17. Sequential Reflection Relations 24 2.18. Signing of Polytopes 24 2.19. Reflection Groups 25 2.20. Reflection group I2(m) 26 2.21. Reflection group An−1 27 2.22. Reflection group Bn 28 2.23. Reflection group Dn 29 2.24. Huffman Polytopes 30 Chapter 3. Planar Graphs 35 3.1. Graph Embeddings 35 3.2. Extended Formulation of T -join Polyhedron 36 3.2.1. Vector Spaces 36 iii iv CONTENTS 3.2.2. Extended Formulation of T -join Polyhedron 37 3.2.3. Construction of Extended Formulation 39 3.3. Extended Formulation of Cut Polytope in Planar Graphs 39 3.3.1. Projecting Linear System 40 3.3.2. Redundant Inequalities 41 3.3.3. Extended Formulation of Cut Polytope 41 3.3.4. Extended Formulation for T -join Polytope and Perfect Matching Polytope 42 3.3.5. Construction of Extended Formulation 42 3.4. Spanning Tree Polytope 42 3.5. Subtour Elimination Polytope 44 3.5.1. Redundant Inequalities 44 3.5.2. Extended Formulation via Spanning Tree Polytope 44 3.5.3. Extended Formulation for Subtour Elimination Polytope 45 Chapter 4. Bounds on General Extended Formulations for Polytopes 47 4.1. Minimal Extended Relaxation 47 4.2. Slack Matrices of Polyhedra 48 4.3. Non-Negative Factorization, Non-Negative Rank 49 4.4. Extended Relaxations from Non-Negative Factorizations 49 4.5. Non-Negative Factorizations from Extended Relaxations 50 4.6. Non-Negative Factorizations, Extensions of Polytopes 51 4.7. Extended Relaxation Problem from Non-Negative Rank Problem 52 4.8. Lattice Embedding 52 4.9. Relaxations of Lattice Embeddings 53 4.10. Rectangle Coverings from Lattice Embeddings 54 4.11. Lattice Embeddings from Rectangle Coverings 55 4.12. Communication Complexity 56 4.13. Upper Bounds on Rectangle Covering Number 57 4.13.1. Matching Polytope 57 4.13.2. Polytopes with Few Vertices on Every Facet 57 4.13.3. Edge Polytopes 58 4.14. Lower Bound on Rectangle Covering 58 4.14.1. Fooling Sets 59 4.14.2. Linear Relaxation 59 4.14.3. Measure of Rectangles 60 4.14.4. Number of Different Sign Patterns in Columns 60 4.15. Rectangle Covering: Graph Point of View 60 4.16. Lower Bounds on Rectangle Covering Number: Rectangle Measures 61 4.17. Lower Bounds on Rectangle Covering Number: Fooling Set 62 4.17.1. Combinatorial Cube 62 4.17.2. Birkhoff Polytope 62 4.17.3. Matching Polytope in Full Bipartite Graph 63 4.18. Lower Bounds on Rectangle Covering Number: Face Counting 63 4.18.1. Permutahedron 63 4.18.2. Huffman Polytope 63 4.18.3. Cardinality Indicating Polytope 64 4.19. Lower Bounds on Rectangle Covering Number: Direct Application 64 Chapter 5. Bounds on Symmetric Extended Formulations of Polytopes 65 5.1. Symmetric Extensions 65 5.2. Symmetric Extended Formulations 68 5.3. Symmetric Section 69 5.4. Examples: Symmetric Extension, Symmetric Section 71 CONTENTS v 5.5. Faces of a Symmetric Extensions 71 5.6. Yannakakis’ Method 72 5.6.1. Action of Group G 72 5.6.2. Action of Group G = S(n) 73 5.6.3. Section Slack Covectors 73 5.7. Matching Polytope 74 5.7.1. Action of Group G = S(n) 75 5.7.2.
Recommended publications
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok

    7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok

    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai

    LINEAR ALGEBRA METHODS in COMBINATORICS László Babai

    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
  • On the Vertices of the D-Dimensional Birkhoff Polytope B)

    On the Vertices of the D-Dimensional Birkhoff Polytope B)

    Discrete Comput Geom (2014) 51:161–170 DOI 10.1007/s00454-013-9554-5 On the Vertices of the d-Dimensional Birkhoff Polytope Nathan Linial · Zur Luria Received: 28 August 2012 / Revised: 6 October 2013 / Accepted: 9 October 2013 / Published online: 7 November 2013 © Springer Science+Business Media New York 2013 Abstract Let us denote by Ωn the Birkhoff polytope of n × n doubly stochastic ma- trices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n × n permutation matrices. Here we consider a higher- (2) dimensional analog of this basic fact. Let Ωn be the polytope which consists of all tristochastic arrays of order n. These are n × n × n arrays with nonnegative entries in (2) which every line sums to 1. What can be said about Ωn ’s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains n − 1 zeros and a single 1 entry. Indeed, every Latin square of order n is (2) avertexofΩn , but as we show, such vertices constitute only a vanishingly small (2) subset of Ωn ’s vertex set. More concretely, we show that the number of vertices of (2) 3 −o(1) Ωn is at least (Ln) 2 , where Ln is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n × n × n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst.
  • The Geometry of the Simplex Method and Applications to the Assignment Problems

    The Geometry of the Simplex Method and Applications to the Assignment Problems

    The Geometry of the Simplex Method and Applications to the Assignment Problems By Rex Cheung SENIOR THESIS BACHELOR OF SCIENCE in MATHEMATICS in the COLLEGE OF LETTERS AND SCIENCE of the UNIVERSITY OF CALIFORNIA, DAVIS Approved: Jes´usA. De Loera March 13, 2011 i ii ABSTRACT. In this senior thesis, we study different applications of linear programming. We first look at different concepts in linear programming. Then we give a study on some properties of polytopes. Afterwards we investigate in the Hirsch Conjecture and explore the properties of Santos' counterexample as well as a perturbation of this prismatoid. We also give a short survey on some attempts in finding more counterexamples using different mathematical tools. Lastly we present an application of linear programming: the assignment problem. We attempt to solve this assignment problem using the Hungarian Algorithm and the Gale-Shapley Algorithm. Along with the algorithms we also give some properties of the Birkhoff Polytope. The results in this thesis are obtained collaboratively with Jonathan Gonzalez, Karla Lanzas, Irene Ramirez, Jing Lin, and Nancy Tafolla. Contents Chapter 1. Introduction to Linear Programming 1 1.1. Definitions and Motivations 1 1.2. Methods for Solving Linear Programs 2 1.3. Diameter and Pivot Bound 8 1.4. Polytopes 9 Chapter 2. Hirsch Conjecture and Santos' Prismatoid 11 2.1. Introduction 11 2.2. Prismatoid and Santos' Counterexample 13 2.3. Properties of Santos' Prismatoid 15 2.4. A Perturbed Version of Santos' Prismatoid 18 2.5. A Search for Counterexamples 22 Chapter 3. An Application of Linear Programming: The Assignment Problem 25 3.1.
  • Centrally Symmetric Polytopes with Many Faces

    Centrally Symmetric Polytopes with Many Faces

    Centrally symmetric polytopes with many faces by Seung Jin Lee A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2013 Doctoral Committee: Professor Alexander I. Barvinok, Chair Professor Thomas Lam Professor John R. Stembridge Professor Martin J. Strauss Professor Roman Vershynin TABLE OF CONTENTS CHAPTER I. Introduction and main results ............................ 1 II. Symmetric moment curve and its neighborliness ................ 6 2.1 Raked trigonometric polynomials . 7 2.2 Roots and multiplicities . 9 2.3 Parametric families of trigonometric polynomials . 14 2.4 Critical arcs . 19 2.5 Neighborliness of the symmetric moment curve . 28 2.6 The limit of neighborliness . 31 2.7 Neighborliness for generalized moment curves . 37 III. Centrally symmetric polytopes with many faces . 41 3.1 Introduction . 41 3.1.1 Cs neighborliness . 41 3.1.2 Antipodal points . 42 3.2 Centrally symmetric polytopes with many edges . 44 3.3 Applications to strict antipodality problems . 49 3.4 Cs polytopes with many faces of given dimension . 52 3.5 Constructing k-neighborly cs polytopes . 56 APPENDICES :::::::::::::::::::::::::::::::::::::::::: 62 BIBLIOGRAPHY ::::::::::::::::::::::::::::::::::::::: 64 ii CHAPTER I Introduction and main results A polytope is the convex hull of a set of finitely many points in Rd. A polytope P ⊂ Rd is centrally symmetric (cs, for short) if P = −P .A convex body is a compact convex set with non-empty interior. A face of a polytope (or a convex body) P can be defined as the intersection of P and a closed halfspace H such that the boundary of H contains no interior point of P .
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm

    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
  • The Alternating Sign Matrix Polytope

    The Alternating Sign Matrix Polytope

    The Alternating Sign Matrix Polytope Jessica Striker [email protected] University of Minnesota The Alternating Sign Matrix Polytope – p. 1/22 What is an alternating sign matrix? Alternating sign matrices (ASMs) are square matrices with the following properties: The Alternating Sign Matrix Polytope – p. 2/22 What is an alternating sign matrix? Alternating sign matrices (ASMs) are square matrices with the following properties: entries ∈{0, 1, −1} the sum of the entries in each row and column equals 1 nonzero entries in each row and column alternate in sign The Alternating Sign Matrix Polytope – p. 2/22 Examples of ASMs n =3 1 0 0 1 0 0 0 1 0 0 10 0 1 0 0 0 1 1 0 0 1 −1 1 0 0 1 0 1 0 0 0 1 0 10 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 The Alternating Sign Matrix Polytope – p. 3/22 Examples of ASMs n =4 0 100 0 1 00 1 −1 0 1 1 −1 10 0 010 0 1 −1 1 0 100 0 0 10 The Alternating Sign Matrix Polytope – p. 4/22 What is a polytope? A polytope can be defined in two equivalent ways ... The Alternating Sign Matrix Polytope – p. 5/22 What is a polytope? A polytope can be defined in two equivalent ways ... ...as the convex hull of a finite set of points The Alternating Sign Matrix Polytope – p. 5/22 What is a polytope? A polytope can be defined in two equivalent ways ..
  • On the Vertex Degrees of the Skeleton of the Matching Polytope of a Graph

    On the Vertex Degrees of the Skeleton of the Matching Polytope of a Graph

    On the vertex degrees of the skeleton of the matching polytope of a graph Nair Abreu ∗ a, Liliana Costa y b, Carlos Henrique do Nascimento z c and Laura Patuzzi x d aEngenharia de Produ¸c~ao,COPPE, Universidade Federal do Rio de Janeiro, Brasil bCol´egioPedro II, Rio de Janeiro, Brasil cInstituto de Ci^enciasExatas, Universidade Federal Fluminense, Volta Redonda, Brasil dInstituto de Matem´atica,Universidade Federal do Rio de Janeiro, Brasil Abstract The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is the skeleton of the matching polytope of G, denoted G(M(G)). Since the number of vertices of G(M(G)) is huge, the structural properties of these graphs have been studied in particular classes. In this paper, for an arbitrary graph G, we obtain a formulae to compute the degree of a vertex of G(M(G)) and prove that the minimum degree of G(M(G)) is equal to the number of edges of G. Also, we identify the vertices of the skeleton with the minimum degree and characterize regular skeletons of the matching polytopes. arXiv:1701.06210v2 [math.CO] 1 Apr 2017 Keywords: graph, matching polytope, degree of matching. ∗Nair Abreu: [email protected]. The work of this author is partially sup- ported by CNPq, PQ-304177/2013-0 and Universal Project-442241/2014-3. yLiliana Costa: [email protected]. The work of this author is partially supported by the Portuguese Foundation for Science and Technology (FCT) by CIDMA -project UID/MAT/04106/2013.
  • The K-Assignment Polytope

    The K-Assignment Polytope

    The k-assignment polytope Jonna Gill ∗ Matematiska institutionen, Linköpings universitet, 581 83 Linköping, Sweden Svante Linusson Matematiska Institutionen, Royal Institute of Technology(KTH), SE-100 44 Stockholm, Sweden Abstract In this paper we study the structure of the k-assignment polytope, whose vertices are the m × n (0,1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. A representation of the faces by certain bipartite graphs is given. This tool is used to describe properties of the polytope, especially a complete description of the cover relation in the face poset of the polytope and an exact expression for the diameter. An ear decomposition of these bipartite graphs is constructed. Key words: Birkhoff polytope, partial matching, face poset, ear decomposition, assignment polytope 1 Introduction The Birkhoff polytope and its properties have been studied from different viewpoints, see e.g. [2,3,4,7]. The Birkhoff polytope Bn has the n × n per- mutation matrices as vertices and is known under many names, such as ‘The polytope of doubly stochastic matrices’ or ‘The assignment polytope’. A natu- ral generalisation of permutation matrices occurring both in optimisation and in theoretical combinatorics is k-assignments. A k-assignment is k entries in ∗ Corresponding author. Telephone number: +46 13282863. Fax number: +46 13100746 Email addresses: [email protected] (Jonna Gill ), [email protected] (Svante Linusson). Preprint submitted to Elsevier Science September 17, 2008 a matrix that are required to be in different rows and columns.
  • Combinatorial Aspects of Convex Polytopes Margaret M

    Combinatorial Aspects of Convex Polytopes Margaret M

    Combinatorial Aspects of Convex Polytopes Margaret M. Bayer1 Department of Mathematics University of Kansas Carl W. Lee2 Department of Mathematics University of Kentucky August 1, 1991 Chapter for Handbook on Convex Geometry P. Gruber and J. Wills, Editors 1Supported in part by NSF grant DMS-8801078. 2Supported in part by NSF grant DMS-8802933, by NSA grant MDA904-89-H-2038, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, NSF-STC88-09648. 1 Definitions and Fundamental Results 3 1.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Faces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.3 Polarity and Duality : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.4 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Shellings 4 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Euler's Relation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Line Shellings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.4 Shellable Simplicial Complexes : : : : : : : : : : : : : : : : : : : 5 2.5 The Dehn-Sommerville Equations : : : : : : : : : : : : : : : : : : 6 2.6 Completely Unimodal Numberings and Orientations : : : : : : : 7 2.7 The Upper Bound Theorem : : : : : : : : : : : : : : : : : : : : : 8 2.8 The Lower Bound Theorem : : : : : : : : : : : : : : : : : : : : : 9 2.9 Constructions Using Shellings : : : : : : : : : : : : :
  • $ K $-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs

    $ K $-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs

    k-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs Rahul Gangopadhyay1 and Saswata Shannigrahi2 1 IIIT Delhi, India [email protected] 2 Saint Petersburg State University, Russia [email protected] Abstract. In this paper, we study the d-dimensional rectilinear drawings d of the complete d-uniform hypergraph K2d. Anshu et al. [Computational Geometry: Theory and Applications, 2017] used Gale transform and Ham- Sandwich theorem to prove that there exist Ω 2d crossing pairs of hy- d peredges in such a drawing of K2d. We improve this lower bound by showing that there exist Ω 2d√d crossing pairs of hyperedges in a d- d dimensional rectilinear drawing of K2d. We also prove the following re- sults. 1. There are Ω 2dd3/2 crossing pairs of hyperedges in a d-dimensional d rectilinear drawing of K2d when its 2d vertices are either not in convex position in Rd or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t + 1)-neighborly for some constant t 1 ≥ independent of d. 2. There are Ω 2dd5/2 crossing pairs of hyperedges in a d-dimensional d rectilinear drawing of K2d when its 2d vertices form the vertices of a d- dimensional convex polytope that is ( d/2 t )-neighborly for some ⌊ ⌋ − ′ constant t 0 independent of d. ′ ≥ Keywords: Rectilinear Crossing Number · Neighborly Polytope · Gale · · · · arXiv:1806.02574v5 [math.CO] 28 Oct 2019 Transform Affine Gale Diagram Balanced Line j-Facet k-Set 1 Introduction d For a sufficiently large positive integer d, let Kn denote the complete d-uniform d hypergraph with n 2d vertices.
  • Recent Progress on Polytopes Günter M. Ziegler∗

    Recent Progress on Polytopes Günter M. Ziegler∗

    Recent Progress on Polytopes G¨unter M. Ziegler∗ Dept. Mathematics 6-1, Technische Universit¨at Berlin 10623 Berlin, Germany [email protected] January 18, 1997 This is a discussion of five very active and important areas of research on the (combinato- rial) theory of (convex) polytopes, with reports about recent progress, and a selection of seven “challenge” problems that I hope to see solved soon: Universality Theorems for polytopes of constant dimension: see Richter-Gebert’s work! Challenge: Can all 3-dimensional polytopes be realized with coordinates of polynomial size? Challenge: Provide a Universality Theorem for simplicial 4-dimensional polytopes. Triangulations and subdivisions of polytopes. Challenge: Decide whether all triangulations on a fixed point set in general position can be connected by bistellar flips. 0/1-polytopes and their combinatorial structure. Challenge: The “0/1 Upper Bound Problem”: Is the maximal number of facets of 0/1- polytopes bounded by an exponential function in the dimension? Neighborly polytopes Explicit constructions and extremal properties. Challenge: Is every polytope a quotient of a neighborly polytope? Monotone paths and the simplex algorithm for linear programming. Challenge: The “Monotone Upper Bound Problem”: What is the maximal number of vertices of a monotone path on a d-dimensional polytope with n facets? Challenge: Is there a polynomial upper bound for the running time of the RANDOM- EDGE simplex algorithm? Disclaimer. This discussion is (solely) concerned with the combinatorial theory of convex polytopes — recent progress, and it is a personal selection of topics, problems and directions that I consider to be interesting. It is meant to be very informal, and cannot provide more than a sketch that hopefully makes you ask for more details and look at some of the references.