Doubly Stochastic Matrices and the Assignment Problem
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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. -
Math 511 Advanced Linear Algebra Spring 2006
MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK 2 x2:1 : 2; 5; 9; 12 x2:3 : 3; 6 x2:4 : 2; 4; 5; 9; 11 Section 2:1: Unitary Matrices Problem 2 If ¸ 2 σ(U) and U 2 Mn is unitary, show that j¸j = 1. Solution. If ¸ 2 σ(U), U 2 Mn is unitary, and Ux = ¸x for x 6= 0, then by Theorem 2:1:4(g), we have kxkCn = kUxkCn = k¸xkCn = j¸jkxkCn , hence j¸j = 1, as desired. Problem 5 Show that the permutation matrices in Mn are orthogonal and that the permutation matrices form a sub- group of the group of real orthogonal matrices. How many different permutation matrices are there in Mn? Solution. By definition, a matrix P 2 Mn is called a permutation matrix if exactly one entry in each row n and column is equal to 1, and all other entries are 0. That is, letting ei 2 C denote the standard basis n th element of C that has a 1 in the i row and zeros elsewhere, and Sn be the set of all permutations on n th elements, then P = [eσ(1) j ¢ ¢ ¢ j eσ(n)] = Pσ for some permutation σ 2 Sn such that σ(k) denotes the k member of σ. Observe that for any σ 2 Sn, and as ½ 1 if i = j eT e = σ(i) σ(j) 0 otherwise for 1 · i · j · n by the definition of ei, we have that 2 3 T T eσ(1)eσ(1) ¢ ¢ ¢ eσ(1)eσ(n) T 6 . -
Chapter Four Determinants
Chapter Four Determinants In the first chapter of this book we considered linear systems and we picked out the special case of systems with the same number of equations as unknowns, those of the form T~x = ~b where T is a square matrix. We noted a distinction between two classes of T ’s. While such systems may have a unique solution or no solutions or infinitely many solutions, if a particular T is associated with a unique solution in any system, such as the homogeneous system ~b = ~0, then T is associated with a unique solution for every ~b. We call such a matrix of coefficients ‘nonsingular’. The other kind of T , where every linear system for which it is the matrix of coefficients has either no solution or infinitely many solutions, we call ‘singular’. Through the second and third chapters the value of this distinction has been a theme. For instance, we now know that nonsingularity of an n£n matrix T is equivalent to each of these: ² a system T~x = ~b has a solution, and that solution is unique; ² Gauss-Jordan reduction of T yields an identity matrix; ² the rows of T form a linearly independent set; ² the columns of T form a basis for Rn; ² any map that T represents is an isomorphism; ² an inverse matrix T ¡1 exists. So when we look at a particular square matrix, the question of whether it is nonsingular is one of the first things that we ask. This chapter develops a formula to determine this. (Since we will restrict the discussion to square matrices, in this chapter we will usually simply say ‘matrix’ in place of ‘square matrix’.) More precisely, we will develop infinitely many formulas, one for 1£1 ma- trices, one for 2£2 matrices, etc. -
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. -
Two-State Systems
1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Alternating Sign Matrices, Extensions and Related Cones
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/311671190 Alternating sign matrices, extensions and related cones Article in Advances in Applied Mathematics · May 2017 DOI: 10.1016/j.aam.2016.12.001 CITATIONS READS 0 29 2 authors: Richard A. Brualdi Geir Dahl University of Wisconsin–Madison University of Oslo 252 PUBLICATIONS 3,815 CITATIONS 102 PUBLICATIONS 1,032 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Combinatorial matrix theory; alternating sign matrices View project All content following this page was uploaded by Geir Dahl on 16 December 2016. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. Alternating sign matrices, extensions and related cones Richard A. Brualdi∗ Geir Dahly December 1, 2016 Abstract An alternating sign matrix, or ASM, is a (0; ±1)-matrix where the nonzero entries in each row and column alternate in sign, and where each row and column sum is 1. We study the convex cone generated by ASMs of order n, called the ASM cone, as well as several related cones and polytopes. Some decomposition results are shown, and we find a minimal Hilbert basis of the ASM cone. The notion of (±1)-doubly stochastic matrices and a generalization of ASMs are introduced and various properties are shown. For instance, we give a new short proof of the linear characterization of the ASM polytope, in fact for a more general polytope. -
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. -
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 .. -
More Gaussian Elimination and Matrix Inversion
Week 7 More Gaussian Elimination and Matrix Inversion 7.1 Opening Remarks 7.1.1 Introduction * View at edX 289 Week 7. More Gaussian Elimination and Matrix Inversion 290 7.1.2 Outline 7.1. Opening Remarks..................................... 289 7.1.1. Introduction..................................... 289 7.1.2. Outline....................................... 290 7.1.3. What You Will Learn................................ 291 7.2. When Gaussian Elimination Breaks Down....................... 292 7.2.1. When Gaussian Elimination Works........................ 292 7.2.2. The Problem.................................... 297 7.2.3. Permutations.................................... 299 7.2.4. Gaussian Elimination with Row Swapping (LU Factorization with Partial Pivoting) 304 7.2.5. When Gaussian Elimination Fails Altogether................... 311 7.3. The Inverse Matrix.................................... 312 7.3.1. Inverse Functions in 1D.............................. 312 7.3.2. Back to Linear Transformations.......................... 312 7.3.3. Simple Examples.................................. 314 7.3.4. More Advanced (but Still Simple) Examples................... 320 7.3.5. Properties...................................... 324 7.4. Enrichment......................................... 325 7.4.1. Library Routines for LU with Partial Pivoting................... 325 7.5. Wrap Up.......................................... 327 7.5.1. Homework..................................... 327 7.5.2. Summary...................................... 327 7.1. Opening -
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 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.