Magic graphs and the faces of the Birkhoff polytope

Magic labelings of graphs are studied in great detail by Stanley and Stewart. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.Comment: 9 page

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

Algebraic Combinatorics of Magic Squares

We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.Comment: Ph.D. Thesi

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method

This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diameter of every $d$-dimensional convex polytope with $n$ facets is bounded above by $n-d$. In particular, we prove a new quadratic upper bound on the diameter of $3$-way axial transportation polytopes defined by $1$-marginals. We also show that the Hirsch Conjecture holds for $p \times 2$ classical transportation polytopes, but that there are infinitely-many Hirsch-sharp classical transportation polytopes. Second, we present new results on subpolytopes of transportation polytopes. We investigate, for example, a non-regular triangulation of a subpolytope of the fourth Birkhoff polytope $B_4$. This implies the existence of non-regular triangulations of all Birkhoff polytopes $B_n$ for $n \geq 4$. We also study certain classes of network flow polytopes and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California, Davis. 183 pages, 49 figure

Alternating sign matrices and polytopes

This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Alternating sign matrices and polytopes

This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Alternating sign matrices and polytopes

This thesis deals with two types of mathematical objects: alternating sign matrices and polytopes. Alternating sign matrices were first defined in 1982 by Mills, Robbins and Rumsey. Since then, alternating sign matrices have led to some very captivating research (with multiple open problems still standing), an outline of which is presented in the opening chapter of this thesis. Convex polytopes are extremely relevant when considering enumerations of certain classes of integer valued matrices. An overview of the relevant properties of convex polytopes is presented, before a connection is made between polytopes and alternating sign matrices: the alternating sign matrix polytope. The vertex set of this new polytope is given, as well as a generalization of standard alternating sign matrices to give higher spin alternating sign matrices. From a result of Ehrhart a result concerning the enumeration of these matrices is obtained, namely, that for fixed size and variable line sum the enumeration is given by a particular polynomial. In Chapter 4, we give results concerning the symmetry classes of the alternating sign matrix polytope and in Chapter 3 we study symmetry classes of the Birkhoff polytope. For this classical polytope we give some new results. In the penultimate chapter, another polytope is defined that is a valid solution set of the transportation problem and for which a particular set of parameters gives the alternating sign matrix polytope. Importantly the transportation polytope is a subset of this new polytope