An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

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

Mini-Workshop: Ehrhart-Quasipolynomials: Algebra, Combinatorics, and Geometry

[no abstract available

Volumes and Integer Points of Multi-Index Transportation Polytopes.

Counting the integer points of transportation polytopes has important applications in statistics for tests of statistical significance, as well as in several applications in combinatorics. In this dissertation, we derive asymptotic formulas for the number of integer and binary integer points in a wide class of multi-index transportation polytopes. A simple closed form approximation is given as the size of the corresponding arrays goes to infinity. A formula for the volume of $4$-index transportation polytopes is also given. We follow the approach of Barvinok and Hartigan to estimate the quantities through a type of local Central Limit Theorem. By carefully estimating eigenvalues and eigenvectors of certain quadratic forms, we are able to prove strong concentration results for the corresponding multivariate Gaussian random variables. We also estimate correlations between linear functions of Gaussian random variables to produce concentration results for the distribution of certain exponential functions. Combined, these techniques allow us to give a complete accounting of the integrals of several functions that are key to estimating the number of integer or binary integer points in multi-index transportation polytopes. As an additional result, we transform a standard concentration of measure on the sphere argument to a concentration result for Gaussian measures whose quadratic forms contain several small eigenvalues, allowing a similar calculation for the volume of multi-index transportation polytopes.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111456/1/dputnins_1.pd

Discrete Geometry

[no abstract available