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we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes.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

Disclosure Analysis for Two-Way Contingency Tables

Ministry of Education, Singapore under its Academic Research Funding Tier 1; SMU Research Offic

A generalization of the integer linear infeasibility problem

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is closely related to non-squarefree elements of Markov bases for sampling contingency tables with given marginals. To study a family of systems with no integer solution, we focus on a commutative semigroup generated by a finite subset of $\Z^d$ and its saturation. An element in the difference of the semigroup and its saturation is called a ``hole''. We show the necessary and sufficient conditions for the finiteness of the set of holes. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the set of holes, the set of non-saturation points, and the set of generators for saturation points. We apply our results to some three- and four-way contingency tables. Then we will discuss the time complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5, Issue 1 (2008) p36-5

Evaluating Theories of Income Dynamics: A Probabilistic Approach

The paper proposes an approach to evaluate hypotheses about transition dynamics when only the distributions at two points in time are observed. Using principles of statistical mechanics, we show how to adjust in the "most probable" way a hypothesis such that it becomes compatible with the observed distributions. This adjustment procedure also allows to test hypotheses in a statistical sense. The test is based on the relative entropy and is equivalent to a likelihood ratio test. We apply our approach to compare the dynamics of the income distribution between men and women in the U.S. using PSID data.income dynamics; large deviation; relative entropy; misspecification