DOCSLIB.ORG

On Some Problems Related to 2-Level Polytopes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Some Problems Related to 2-Level Polytopes On some problems related to 2-level polytopes Thèse n. 9025 2018 présenté le 22 Octobre 2018 à la Faculté des Sciences de Base chaire d'optimisation discrète programme doctoral en mathématiques École Polytechnique Fédérale de Lausanne pour l'obtention du grade de Docteur ès Sciences par Manuel Francesco Aprile acceptée sur proposition du jury: Prof Hess Bellwald Kathryn, président du jury Prof Eisenbrand Friedrich, directeur de thèse Prof Faenza Yuri, codirecteur de thèse Prof Svensson Ola, rapporteur Prof Fiorini Samuel, rapporteur Prof Weltge Stefan, rapporteur Lausanne, EPFL, 2018 "You must go from wish to wish. What you don’t wish for will always be beyond your reach." — Michael Ende, The Neverending Story. To my brother Stefano, to whom I dedicated my Bachelor thesis too. He is now six years old and he has not read it yet. Maybe one day? Acknowledgements My PhD has been an incredible journey of discovery and personal growth. There were mo- ments of frustration and doubt, when research just wouldn’t work, even depression, but there was also excitement, enjoyment, and satisfaction for my progresses. Oh, there was hard work too. A whole lot of people contributed to making this experience so amazing. First, I would like to thank my advisor Fritz Eisenbrand for his support, the trust he put in me, and for letting me do research in total freedom. He allowed me to go to quite some conferences and workshops, where I could expand my knowledge and professional network, as well as enjoying some traveling. A big “grazie” goes to Yuri Faenza, my co-advisor: in a moment when I was not sure about the direction of my research, he gave a talk about 2-level polytopes, immediately catching my interest, and, a few days after, he was introducing me to extended formulations with an improvised crash course. Ironically, that day he explained to me Yannakakis’ protocol, which became the core of the last chapter of my thesis. He invested many hours working with me, taught me so much, he even hosted me at his place in New York (thanks to his wife Fanny too!): it goes without saying that this thesis would not have been possible without him. I am indebted to all researchers I met, from which I gathered interesting conversations, inspira- tion, precious pieces of advice. A complete list would be too long, but I would like to mention Michele Conforti, Marco Di Summa, Gwenaël Joret, Volker Kaibel, Monaldo Mastrolilli, Janos Pach. In particular I thank Kathryn Hess Bellwald, Ola Svensson, Stefan Weltge for being part of my PhD jury. Heartfelt thanks to Samuel Fiorini for inviting me to Bruxelles and for teaching me so much in so little time. Thanks to Tony Huynh and Marco Macchia, it was great to work and go for beers with you. I look forward to repeating both experiences. Thanks to Alfonso Cevallos, best office mate ever, for all the adventures, discussions, and for solving problems that I could not solve. Thanks to each member of DISOPT group, in particular to Igor Malinovi´c and Christoph Hunkenschröder for their valuable friendship and all the fun activities we did together, and to the members of DCG group too. We had quite a good time! I am also very grateful to Jocelyne and Natacha for their kindness and precious administrative support. I also had the luck to make many friends in the math department and beyond, among which I would like to mention Riccardo (worst housemate!), Clelia, Francesco, Gennaro, Jessica, Arwa, Lydia, Thomas, Chiara, Mauro. They might not have directly contributed to my work, but definitely made my time off much more enjoyable. Finally, the support and the love I had from Sicily was very important to me, especially during difficult moments. Thanks to all my Sicilian friends. Thanks to Carla, who made the last part of my PhD way better than the first in so many aspects: trying to explain my research to you v Acknowledgements was so fun that I plan to do it again and again, you have been warned. I am much grateful to all my family and in particular to my parents, for being extremely supportive and for encouraging me to pursue my dreams with no limits but my ambition, even though this meant to go far from home. Grazie. Lausanne, 17 September 2018 vi Abstract In this thesis we investigate a number of problems related to 2-level polytopes, in particular regarding their combinatorial structure and extension complexity. 2-level polytopes have been introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems: these include connec- tion with communication complexity and the separation between linear and semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter we introduce the main concepts that will be used through the thesis and we motivate our interest in 2-level polytopes. In the second chapter we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the third chapter we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain improved lower and upper bounds. In the fourth chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fifth chapter we address the problem of explicitly obtaining small size extended formu- lations whose existence is guaranteed by communication protocols. In particular we give an output-efficient algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols. We then conclude the thesis outlining the main open questions that stem from our work. Keywords: Polytopes, polyhedral combinatorics, 2-level, extension complexity, vertices, facets, slack matrix. vii Sommario In questa tesi vengono trattati diversi problemi sui politopi "2-level", in particolare sulla loro struttura combinatoria e complessità di estensione. Tali politopi sono una generalizzazione di politopi che derivano dagli insiemi indipendenti nei grafi perfetti, e, nonostante la loro struttura apparentemente semplice, sono al centro di molti problemi aperti che spaziano dalla complessità computazionale alla programmazione semidefinita. La complessità di estensione di un politopo P è una misura della complessità nel rappresentare P: è la minima dimensione di una formulazione estesa di P, che a sua volta è una descrizione lineare di un poliedro di cui P è la proiezione. Nel primo capitolo vengono introdotti i politopi 2-level e i concetti principali che verran- no usati nella tesi, e vengono descritte le principali motivazioni dell’interesse verso questi politopi. Nel secondo capitolo vengono esaminate diverse classi di politopi 2-level che appaiono in contesti combinatori, e viene provata una relazione tra il numero di faccette e di vertici di tali politopi. Congetturiamo che tale relazione valga per tutti i politopi 2-level. Le dimostrazioni vengono ottenute tramite una migliore comprensione della struttura combinatoria di tali politopi, che a volte porta a risultati interessanti a prescindere dalla congettura. Nel terzo capitolo studiamo la complessità di estensione di una particolare classe di politopi 2-level, derivante dagli insiemi indipendenti dei grafi bipartiti, di cui miglioriamo il limite inferiore e superiore. Nel quarto capitolo studiamo le matrici di slack dei politopi 2-level, importanti oggetti combi- natori collegati alla complessità di estensione, definiamo operazioni su tali matrici e diamo algoritmi per il seguente problema: data una matrice, determinare se è una matrice di slack di una certa classe di politopi 2-level. Nel quinto capitolo affrontiamo il problema di ottenere formulazioni estese compatte ed espli- cite, quando l’esistenza di tali formulazioni è dimostrata tramite protocolli di comunicazione. In particolare diamo un algoritmo per ottenere una formulazione estesa del politopo degli insiemi indipendenti dei grafi perfetti, rendendo costruttivo un noto risultato di Yannakakis. Il risultato è abbastanza generale da essere applicabile a tutti i protocolli deterministici. La tesi si conclude con una discussione delle principali direzioni di ricerca che scaturiscono dal nostro lavoro. Parole chiave: politopi, 2-level, formulazioni estese, vertici, faccette, matrice di slack. ix Contents Acknowledgementsv Abstract (English/Italiano) vii 1 Introduction 1 1.1 Preliminaries.......................................6 2 On vertices and facets of 2-level polytopes arising in combinatorial settings7 2.1 Introduction........................................7 2.2 Basics............................................9 2.2.1 Hanner and Birkhoff polytopes.........................9 2.3 Graphical 2-Level Polytopes............................... 10 2.3.1 Stable set polytopes of perfect graphs..................... 11 2.3.2 Hansen polytopes................................. 12 2.3.3 Min up/down polytopes............................. 12 2.3.4 Polytopes coming from posets......................... 14 2.3.5 Stable matching polytopes........................... 16 2.4 2-Level Matroid Base Polytopes............................. 19 2.4.1 2-level matroid polytopes and Conjecture 2.1................ 20 2.4.2 Flacets of 2-sums................................. 24 2.4.3 Linear Description of 2-Level Matroid Base Polytopes..........
Load more

Recommended publications
  • The Hypersimplex Canonical Forms and the Momentum Amplituhedron-Like Logarithmic Forms
    The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms TomaszLukowski, 1 and Jonah Stalknecht1 1Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, Hertfordshire, AL10 9AB, United Kingdom E-mail: [email protected], [email protected] Abstract: In this paper we provide a formula for the canonical differential form of the hypersimplex ∆k;n for all n and k. We also study the generalization of the momentum ampli- tuhedron Mn;k to m = 2, and we conclude that the existing definition does not possess the desired properties. Nevertheless, we find interesting momentum amplituhedron-like logarith- mic differential forms in the m = 2 version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms. arXiv:2107.07520v1 [hep-th] 15 Jul 2021 Contents 1 Introduction 1 2 Hypersimplex2 2.1 Definitions3 2.2 Hypersimplex canonical forms4 3 Momentum amplituhedron9 3.1 Definition of m = 2 momentum amplituhedron 10 3.2 Momentum amplituhedron-like logarithmic forms 11 3.3 Geometry 12 4 Summary and outlook 15 A Definition of positive geometry and push-forward 15 1 Introduction Geometry has always played an essential role in physics, and it continues to be crucial in many recently developed branches of theoretical and high-energy physics. In recent years, this statement has been supported by the introduction of positive geometries [1] that encode a variety of observables in Quantum Field Theories [2{5], and beyond [6{8], see [9] for a com- prehensive review. These recent advances have also renewed the interest in well-established and very well-studied geometric objects, allowing us to look at them in a completely new way.
    [Show full text]
  • Two Poset Polytopes
    Discrete Comput Geom 1:9-23 (1986) G eometrv)i.~.reh, ~ ( :*mllmlati~ml © l~fi $1~ter-Vtrlq New Yorklu¢. t¢ Two Poset Polytopes Richard P. Stanley* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. Two convex polytopes, called the order polytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart poly- nomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer map" then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities. I. The Order Polytope Our aim is to investigate two convex polytopes associated with a finite partially ordered set (poset) P. The first of these, which we call the "order polytope" and denote by O(P), has been the subject of considerable scrutiny, both explicit and implicit, Much of what we say about the order polytope will be essentially a review of well-known results, albeit ones scattered throughout the literature, sometimes in a rather obscure form. The second polytope, called the "chain polytope" and denoted if(P), seems never to have been previously considered per se. It is a special case of the vertex-packing polytope of a graph (see Section 2) but has many special properties not in general valid or meaningful for graphs.
    [Show full text]
  • 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.
    [Show full text]
  • Double Posets and Real Invariant Varieties Dissertation
    Fachbereich Mathematik und Informatik der Freien Universität Berlin Double posets and real invariant varieties Two interactions between combinatorics and geometry Dissertation eingereicht von Tobias Friedl Berlin 2017 Advisor and first reviewer: Prof. Dr. Raman Sanyal Second reviewer: Prof. Dr. Francisco Santos Third reviewer: Priv.-Doz. Dr. Christian Stump Date of the defense: May 19, 2017 Acknowledgements My deepest thanks go to my advisor Raman Sanyal. A PhD-student can only hope for an advisor who is as dedicated and enthusiastic about mathematics as you are. Thank you for getting your hands dirty and spending many hours in front of the blackboard teaching me how to do research. I spent the last years in the amazing and inspiring work group directed by Günter Ziegler at FU Berlin. Thank you for providing such a welcoming and challenging work environment. I really enjoyed the time with my friends and colleagues at the "villa", most of all Francesco Grande, Katharina Jochemko, Katy Beeler, Albert Haase, Lauri Loiskekoski, Philip Brinkmann, Nevena Palić and Jean-Philippe Labbé. Thanks also to Elke Pose, who provides valuable support and keeps bureaucracy at a low level. I want to thank my coauthor Cordian Riener for many fruitful mathematical discussions and for an interesting and enjoyable week of research in Helsinki at Aalto University. Moreover, I’d like to express my gratitude towards my coauthor and friend Tom Chappell. Thanks to the reviewers Paco Santos and Christian Stump for their helpful com- ments and to Christian additionally for many discussions regarding the combinatorics of reflection groups and posets throughout the last years.
    [Show full text]
  • Arxiv:1902.07301V3 [Math.CO] 13 Feb 2020 Which N H Itiuinwhere Distribution the and of Ojcuefo 8]Wihhsntbe Eovd H Oere Sole the Resolved
    MINUSCULE DOPPELGANGERS,¨ THE COINCIDENTAL DOWN-DEGREE EXPECTATIONS PROPERTY, AND ROWMOTION SAM HOPKINS Abstract. We relate Reiner, Tenner, and Yong’s coincidental down-degree expec- tations (CDE) property of posets to the minuscule doppelg¨anger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelg¨anger pairs behave “as if” they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelg¨anger pairs pretending to have isomor- phic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelg¨anger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type). 1. Introduction Let P be a finite poset. The down-degree of p ∈ P is the number of elements of P which p covers. Consider two probability distributions on P : the uniform distribution; and the distribution where p ∈ P occurs proportional to the number of maximal chains of P containing p. We say that P has the coincidental down-degree expectations (CDE) property if the expected value of the down-degree statistic is the same for these two distributions.
    [Show full text]
  • 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.
    [Show full text]
  • Polyhedral Combinatorics, Complexity & Algorithms
    POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS By FOAD MAHDAVI PAJOUH Bachelor of Science in Industrial Engineering Sharif University of Technology Tehran, Iran 2004 Master of Science in Industrial Engineering Tarbiat Modares University Tehran, Iran 2006 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2012 COPYRIGHT c By FOAD MAHDAVI PAJOUH July, 2012 POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS Dissertation Approved: Dr. Balabhaskar Balasundaram Dissertation Advisor Dr. Ricki Ingalls Dr. Manjunath Kamath Dr. Ramesh Sharda Dr. Sheryl Tucker Dean of the Graduate College iii Dedicated to my parents iv ACKNOWLEDGMENTS I would like to wholeheartedly express my sincere gratitude to my advisor, Dr. Balabhaskar Balasundaram, for his continuous support of my Ph.D. study, for his patience, enthusiasm, motivation, and valuable knowledge. It has been a great honor for me to be his first Ph.D. student and I could not have imagined having a better advisor for my Ph.D. research and role model for my future academic career. He has not only been a great mentor during the course of my Ph.D. and professional development but also a real friend and colleague. I hope that I can in turn pass on the research values and the dreams that he has given to me, to my future students. I wish to express my warm and sincere thanks to my wonderful committee mem- bers, Dr. Ricki Ingalls, Dr. Manjunath Kamath and Dr. Ramesh Sharda for their support, guidance and helpful suggestions throughout my Ph.D.
    [Show full text]
  • Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux Dorian Croitoru
    Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux by Dorian Croitoru M.A., University of Pittsburgh, 2005 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2010 © Dorian Croitoru, 2010. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author......................................... Department of Mathematics July 21, 2010 Certifiedby...................................... Alexander Postnikov Associate Professor of Applied Mathematics Thesis Supervisor Accepted by . Bjorn Poonen Chairman, Department Committee on Graduate Students Mixed Volumes of Hypersimplices, Root Systems and Shifted Young Tableaux by Dorian Croitoru Submitted to the Department of Mathematics on July 21, 2010 in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis consists of two parts. In the first part, we start by investigating the classical permutohedra as Minkowski sums of the hypersimplices. Their volumes can be expressed as polynomials whose coefficients { the mixed Eulerian numbers { are given by the mixed volumes of the hypersimplices. We build upon results of Postnikov and derive various recursive and combinatorial formulas for the mixed Eulerian numbers. We generalize these results to arbitrary root systems Φ, and obtain cyclic, recursive and combinatorial formulas for the volumes of the weight polytopes (Φ-analogues of permutohedra) as well as the mixed Φ-Eulerian numbers. These formulas involve Cartan matrices and weighted paths in Dynkin diagrams, and thus enable us to extend the theory of mixed Eulerian numbers to arbitrary matrices whose principal minors are invertible.
    [Show full text]
  • Topics in Algorithmic, Enumerative and Geometric Combinatorics
    Thesis for the Degree of Doctor of Philosophy Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij Division of Mathematics Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg G¨oteborg, Sweden 2012 Topics in algorithmic, enumerative and geometric combinatorics Ragnar Freij ISBN 978-91-7385-668-3 c Ragnar Freij, 2012. Doktorsavhandlingar vid Chalmers Tekniska H¨ogskola Ny serie Nr 3349 ISSN 0346-718X Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GOTEBORG,¨ Sweden Phone: +46 (0)31-772 10 00 [email protected] Printed in G¨oteborg, Sweden, 2012 Topics in algorithmic, enumerative and geometric com- binatorics Ragnar Freij ABSTRACT This thesis presents five papers, studying enumerative and extremal problems on combinatorial structures. The first paper studies Forman’s discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S2 × Sn−2-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties. In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning technique, the number of nonempty faces is counted, and in particular we confirm Kalai’s 3d-conjecture for such polytopes. Further- more, a characterization of exactly which Hansen polytopes are also Hanner polytopes is given.
    [Show full text]
  • The $ H^* $-Polynomial of the Order Polytope of the Zig-Zag Poset
    THE h∗-POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET JANE IVY COONS AND SETH SULLIVANT Abstract. We describe a family of shellings for the canonical tri- angulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numer- ator of the Ehrhart series of this order polytope in terms of the swap statistic on alternating permutations. 1. Introduction and Preliminaries The zig-zag poset Zn on ground set {z1,...,zn} is the poset with exactly the cover relations z1 < z2 > z3 < z4 > . That is, this partial order satisfies z2i−1 < z2i and z2i > z2i+1 for all i between n−1 1 and ⌊ 2 ⌋. The order polytope of Zn, denoted O(Zn) is the set n of all n-tuples (x1,...,xn) ∈ R that satisfy 0 ≤ xi ≤ 1 for all i and xi ≤ xj whenever zi < zj in Zn. In this paper, we introduce the “swap” permutation statistic on alternating permutations to give a new combinatorial interpretation of the numerator of the Ehrhart series of O(Zn). We began studying this problem in relation to combinatorial proper- ties of the Cavender-Farris-Neyman model with a molecular clock (or CFN-MC model) from mathematical phylogenetics [4]. We were inter- ested in the polytope associated to the toric variety obtained by ap- plying the discrete Fourier transform to the Cavender-Farris-Neyman arXiv:1901.07443v2 [math.CO] 1 Apr 2020 model with a molecular clock on a given rooted binary phylogenetic tree. We call this polytope the CFN-MC polytope.
    [Show full text]
  • 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.
    [Show full text]
  • 3. Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics
    Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory lectures on linear programming and polyhedral combinatorics. Definition 3.1 A halfspace in Rn is a set of the form fx 2 Rn : aT x ≤ bg for some vector a 2 Rn and b 2 R. Definition 3.2 A polyhedron is the intersection of finitely many halfspaces: P = fx 2 Rn : Ax ≤ bg. Definition 3.3 A polytope is a bounded polyhedron. n n−1 Definition 3.4 If P is a polyhedron in R , the projection Pk ⊆ R of P is defined as fy = (x1; x2; ··· ; xk−1; xk+1; ··· ; xn): x 2 P for some xkg. This is a special case of a projection onto a linear space (here, we consider only coordinate projection). By repeatedly projecting, we can eliminate any subset of coordinates. We claim that Pk is also a polyhedron and this can be proved by giving an explicit description of Pk in terms of linear inequalities. For this purpose, one uses Fourier-Motzkin elimination. Let P = fx : Ax ≤ bg and let • S+ = fi : aik > 0g, • S− = fi : aik < 0g, • S0 = fi : aik = 0g. T Clearly, any element in Pk must satisfy the inequality ai x ≤ bi for all i 2 S0 (these inequal- ities do not involve xk). Similarly, we can take a linear combination of an inequality in S+ and one in S− to eliminate the coefficient of xk. This shows that the inequalities: ! ! X X aik aljxj − alk aijxj ≤ aikbl − alkbi (1) j j for i 2 S+ and l 2 S− are satisfied by all elements of Pk.
    [Show full text]