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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 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. -
Combinatorial and Discrete Problems in Convex Geometry
COMBINATORIAL AND DISCRETE PROBLEMS IN CONVEX GEOMETRY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Matthew R. Alexander September, 2017 Dissertation written by Matthew R. Alexander B.S., Youngstown State University, 2011 M.A., Kent State University, 2013 Ph.D., Kent State University, 2017 Approved by Dr. Artem Zvavitch , Co-Chair, Doctoral Dissertation Committee Dr. Matthieu Fradelizi , Co-Chair Dr. Dmitry Ryabogin , Members, Doctoral Dissertation Committee Dr. Fedor Nazarov Dr. Feodor F. Dragan Dr. Jonathan Maletic Accepted by Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS LIST OF FIGURES . vi ACKNOWLEDGEMENTS ........................................ vii NOTATION ................................................. viii I Introduction 1 1 This Thesis . .2 2 Preliminaries . .4 2.1 Convex Geometry . .4 2.2 Basic functions and their properties . .5 2.3 Classical theorems . .6 2.4 Polarity . .8 2.5 Volume Product . .8 II The Discrete Slicing Problem 11 3 The Geometry of Numbers . 12 3.1 Introduction . 12 3.2 Lattices . 12 3.3 Minkowski's Theorems . 14 3.4 The Ehrhart Polynomial . 15 4 Geometric Tomography . 17 4.1 Introduction . 17 iii 4.2 Projections of sets . 18 4.3 Sections of sets . 19 4.4 The Isomorphic Busemann-Petty Problem . 20 5 The Discrete Slicing Problem . 22 5.1 Introduction . 22 5.2 Solution in Z2 ............................................. 23 5.3 Approach via Discrete F. John Theorem . 23 5.4 Solution using known inequalities . 26 5.5 Solution for Unconditional Bodies . 27 5.6 Discrete Brunn's Theorem . -
Classification of Ehrhart Polynomials of Integral Simplices Akihiro Higashitani
Classification of Ehrhart polynomials of integral simplices Akihiro Higashitani To cite this version: Akihiro Higashitani. Classification of Ehrhart polynomials of integral simplices. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.587-594. hal-01283113 HAL Id: hal-01283113 https://hal.archives-ouvertes.fr/hal-01283113 Submitted on 5 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 587–594 Classification of Ehrhart polynomials of integral simplices Akihiro Higashitani y Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract. Let δ(P) = (δ0; δ1; : : : ; δd) be the δ-vector of an integral convex polytope P of dimension d. First, by Pd using two well-known inequalities on δ-vectors, we classify the possible δ-vectors with i=0 δi ≤ 3. Moreover, by Pd means of Hermite normal forms of square matrices, we also classify the possible δ-vectors with i=0 δi = 4. In Pd Pd addition, for i=0 δi ≥ 5, we characterize the δ-vectors of integral simplices when i=0 δi is prime. -
Unimodular Triangulations of Dilated 3-Polytopes
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 74 (2013), vyp. 2 2013, Pages 293–311 S 0077-1554(2014)00220-X Article electronically published on April 9, 2014 UNIMODULAR TRIANGULATIONS OF DILATED 3-POLYTOPES F. SANTOS AND G. M. ZIEGLER Abstract. A seminal result in the theory of toric varieties, by Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope P there is a positive integer k such that the dilated polytope kP has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that k = 4 works for every polytope. But this does not imply that every k>4 works as well. We here study the values of k for which the result holds, showing that: (1) It contains all composite numbers. (2) It is an additive semigroup. These two properties imply that the only values of k that may not work (besides 1 and 2, which are known not to work) are k ∈{3, 5, 7, 11}.Withanad-hocconstruc- tion we show that k =7andk = 11 also work, except in this case the triangulation cannot be guaranteed to be “standard” in the boundary. All in all, the only open cases are k =3andk =5. § 1. Introduction Let Λ ⊂ Rd be an affine lattice. A lattice polytope (or an integral polytope [4]) is a polytope P with all its vertices in Λ. We are particularly interested in lattice (full- dimensional) simplices. The vertices of a lattice simplex Δ are an affine basis for Rd, hence they induce a sublattice ΛΔ of Λ of index equal to the normalized volume of Δ with respect to Λ. -
Algorithmic System Design Under Consideration of Dynamic Processes
Algorithmic System Design under Consideration of Dynamic Processes Vom Fachbereich Maschinenbau an der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation vorgelegt von Dipl.-Phys. Lena Charlotte Altherr aus Karlsruhe Berichterstatter: Prof. Dr.-Ing. Peter F. Pelz Mitberichterstatter: Prof. Dr. rer. nat. Ulf Lorenz Tag der Einreichung: 19.04.2016 Tag der m¨undlichen Pr¨ufung: 24.05.2016 Darmstadt 2016 D 17 Vorwort des Herausgebers Kontext Die Produkt- und Systementwicklung hat die Aufgabe technische Systeme so zu gestalten, dass eine gewünschte Systemfunktion erfüllt wird. Mögliche System- funktionen sind z.B. Schwingungen zu dämpfen, Wasser in einem Siedlungsgebiet zu verteilen oder die Kühlung eines Rechenzentrums. Wir Ingenieure reduzieren dabei die Komplexität eines Systems, indem wir dieses gedanklich in überschaubare Baugruppen oder Komponenten zerlegen und diese für sich in Robustheit und Effizienz verbessern. In der Kriegsführung wurde dieses Prinzip bereits 500 v. Chr. als „Teile und herrsche Prinzip“ durch Meister Sun in seinem Buch „Die Kunst der Kriegsführung“ beschrieben. Das Denken in Schnitten ist wesentlich für das Verständnis von Systemen: „Das wichtigste Werkzeug des Ingenieurs ist die Schere“. Das Zusammenwirken der Komponenten führt anschließend zu der gewünschten Systemfunktion. Während die Funktion eines technischen Systems i.d.R. nicht verhan- delbar ist, ist jedoch verhandelbar mit welchem Aufwand diese erfüllt wird und mit welcher Verfügbarkeit sie gewährleistet wird. Aufwand und Verfügbarkeit sind dabei gegensätzlich. Der Aufwand bemisst z.B. die Emission von Kohlendioxid, den Energieverbrauch, den Materialverbrauch, … die „total cost of ownership“. Die Verfügbarkeit bemisst die Ausfallzeiten, Lebensdauer oder Laufleistung. Die Gesell- schaft stellt sich zunehmend die Frage, wie eine Funktion bei minimalem Aufwand und maximaler Verfügbarkeit realisiert werden kann. -
54 OP08 Abstracts
54 OP08 Abstracts CP1 Dept. of Mathematics Improving Ultimate Convergence of An Aug- [email protected] mented Lagrangian Method Optimization methods that employ the classical Powell- CP1 Hestenes-Rockafellar Augmented Lagrangian are useful A Second-Derivative SQP Method for Noncon- tools for solving Nonlinear Programming problems. Their vex Optimization Problems with Inequality Con- reputation decreased in the last ten years due to the com- straints parative success of Interior-Point Newtonian algorithms, which are asymptotically faster. In the present research We consider a second-derivative 1 sequential quadratic a combination of both approaches is evaluated. The idea programming trust-region method for large-scale nonlin- is to produce a competitive method, being more robust ear non-convex optimization problems with inequality con- and efficient than its ”pure” counterparts for critical prob- straints. Trial steps are composed of two components; a lems. Moreover, an additional hybrid algorithm is defined, Cauchy globalization step and an SQP correction step. A in which the Interior Point method is replaced by the New- single linear artificial constraint is incorporated that en- tonian resolution of a KKT system identified by the Aug- sures non-accent in the SQP correction step, thus ”guiding” mented Lagrangian algorithm. the algorithm through areas of indefiniteness. A salient feature of our approach is feasibility of all subproblems. Ernesto G. Birgin IME-USP Daniel Robinson Department of Computer Science Oxford University [email protected] -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Kombinatorische Optimierung – Blatt 1
Prof. Dr. Volker Kaibel M.Sc. Benjamin Peters Wintersemester 2016/2017 Kombinatorische Optimierung { Blatt 1 www.math.uni-magdeburg.de/institute/imo/teaching/wise16/kombopt/ Pr¨asentation in der Ubung¨ am 20.10.2016 Aufgabe 1 Betrachte das Hamilton-Weg-Problem: Gegeben sei ein Digraph D = (V; A) sowie ver- schiedene Knoten s; t ∈ V . Ein Hamilton-Weg von s nach t ist ein s-t-Weg, der jeden Knoten in V genau einmal besucht. Das Problem besteht darin, zu entscheiden, ob ein Hamilton-Weg von s nach t existiert. Wir nehmen nun an, wir h¨atten einen polynomiellen Algorithmus zur L¨osung des Kurzeste-¨ Wege Problems fur¨ beliebige Bogenl¨angen. Konstruiere damit einen polynomiellen Algo- rithmus fur¨ das Hamilton-Weg-Problem. Aufgabe 2 Der folgende Graph abstrahiert ein Straßennetz. Dabei geben die Kantengewichte die (von einander unabh¨angigen) Wahrscheinlichkeiten an, bei Benutzung der Straßen zu verunfallen. Bestimme den sichersten Weg von s nach t durch Aufstellen und L¨osen eines geeignetes Kurzeste-Wege-Problems.¨ 2% 2 5% 5% 3% 4 1% 5 2% 3% 6 t 6% s 4% 2% 1% 2% 7 2% 1% 3 Aufgabe 3 Lesen Sie den Artikel The Year Combinatorics Blossomed\ (erschienen 2015 im Beijing " Intelligencer, Springer) von William Cook, Martin Gr¨otschel und Alexander Schrijver. S. 1/7 {:::a äi, stq: (: W illianr Cook Lh t ivers it1' o-l' |'\'at ttlo t), dt1 t t Ll il Martin Grötschel ZtLse lnstitrttt; orttl'l-U Ilt:rlin, ()trrnnt1, Alexander Scfu ijver C\\/ I nul Ltn ivtrsi !)' o-l' Anrstt rtlan, |ietherlattds The Year Combinatorics Blossomed One summer in the mid 1980s, Jack Edmonds stopped Much has been written about linear programming, by the Research Institute for Discrete Mathematics including several hundred texts bearing the title. -
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. -
Triangulations of Integral Polytopes, Examples and Problems
数理解析研究所講究録 955 巻 1996 年 133-144 133 Triangulations of integral polytopes, examples and problems Jean-Michel KANTOR We are interested in polytopes in real space of arbitrary dimension, having vertices with integral co- ordinates: integral polytopes. The recent increase of interest for the study of these $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}$ topes and their triangulations has various motivations; let us mention the main ones: . the beautiful theory of toric varieties has built a bridge between algebraic geometry and the combina- torics of these integral polytopes [12]. Triangulations of cones and polytopes occur naturally for example in problems of existence of crepant resolution of singularities $[1,5]$ . The work of the school of $\mathrm{I}.\mathrm{M}$ . Gelfand on secondary polytopes gives a new insight on triangulations, with applications to algebraic geometry and group theory [13]. In statistical physics, random tilings lead to some interesting problems dealing with triangulations of order polytopes $[6,29]$ . With these motivations in mind, we introduce new tools: Generalizations of the Ehrhart polynomial (counting points “modulo congruence”), discrete length between integral points (and studying the geometry associated to it), arithmetic Euler-Poincar\'e formula which gives, in dimension 3, the Ehrhart polynomial in terms of the $\mathrm{f}$-vector of a minimal triangulation of the polytope (Theorem 7). Finally, let us mention the results in dimension 2 of the late Peter Greenberg, they led us to the study of “Arithmetical $\mathrm{P}\mathrm{L}$-topology” which, we believe with M. Gromov, D. Sullivan, and P. Vogel, has not yet revealed all its beauties. -
Matroid Partitioning Algorithm Described in the Paper Here with Ray’S Interest in Doing Ev- Erything Possible by Using Network flow Methods
Chapter 7 Matroid Partition Jack Edmonds Introduction by Jack Edmonds This article, “Matroid Partition”, which first appeared in the book edited by George Dantzig and Pete Veinott, is important to me for many reasons: First for per- sonal memories of my mentors, Alan J. Goldman, George Dantzig, and Al Tucker. Second, for memories of close friends, as well as mentors, Al Lehman, Ray Fulker- son, and Alan Hoffman. Third, for memories of Pete Veinott, who, many years after he invited and published the present paper, became a closest friend. And, finally, for memories of how my mixed-blessing obsession with good characterizations and good algorithms developed. Alan Goldman was my boss at the National Bureau of Standards in Washington, D.C., now the National Institutes of Science and Technology, in the suburbs. He meticulously vetted all of my math including this paper, and I would not have been a math researcher at all if he had not encouraged it when I was a university drop-out trying to support a baby and stay-at-home teenage wife. His mentor at Princeton, Al Tucker, through him of course, invited me with my child and wife to be one of the three junior participants in a 1963 Summer of Combinatorics at the Rand Corporation in California, across the road from Muscle Beach. The Bureau chiefs would not approve this so I quit my job at the Bureau so that I could attend. At the end of the summer Alan hired me back with a big raise. Dantzig was and still is the only historically towering person I have known.