Jesús A. De Loera

Effective lattice point counting in rational convex polytopes

Abstract This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE , a computer package for lattice point enumeration which contains the first implementation of A. Barvinok’s algorithm (Math. Oper. Res. 19 (1994) 769). We report on computational experiments with multiway contingency tables, knapsack type problems, rational polygons, and flow polytopes. We prove that these kinds of symbolic–algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting. Using LattE , we have also computed new formulas of Ehrhart (quasi-)polynomials for interesting families of polytopes (hypersimplices, truncated cubes, etc). We end with a survey of other “algebraic–analytic” algorithms, including a “homogeneous” variation of Barvinok’s algorithm which is very fast when the number of facet-defining inequalities is much smaller compared to the number of vertices.

Markov bases of three-way tables are arbitrarily complicated

We show the following two universality statements on the entry-ranges and Markov bases of spaces of 3-way contingency tables with fixed 2-margins: (1) For any finite set D of nonnegative integers, there are r,c, and 2-margins for (r,c,3)-tables such that the set of values occurring in a fixed entry in all possible tables with these margins is D. (2) For any integer n-vector d, there are r,c such that any Markov basis for (r,c,3)-tables with fixed 2-margins must contain an element whose restriction to some n entries is d. In particular, the degree and support of elements in the minimal Markov bases when r and c vary can be arbitrarily large, in striking contrast with the case for 1-margined tables in any dimension and any format and with 2-margined (r,c,h)-tables with both c,h fixed. These results have implications for confidential statistical data disclosure control. Specifically, they demonstrate that the entry-range of 2-margined 3-tables can contain arbitrary gaps, suggesting that even if the smallest and largest possible values of an entry are far apart, the disclosure of such margins may be insecure. Thus, the behavior of sensitive data under disclosure of aggregated data is far from what has been so far believed. Our results therefore call for the re-examination of aggregation and disclosure practices and for further research on the issues exposed herein. Our constructions also provides a powerful automatic tool in constructing concrete examples, such as the possibly smallest 2-margins for (6, 4, 3)-tables with entry-range containing a gap.

N-fold integer programming

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed n-fold integer programming problems are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and the so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the d-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting-stock problem.

How to Integrate a Polynomial over a Simplex

This paper starts by settling the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods. 1.

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Gröbner basis for the associated toric idealK(Kn). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofKn. Forn≥6 the number of distinct initial ideals ofI(Kn) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(Kn) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Gröbner basis ofI(Kn) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

The many aspects of counting lattice points in polytopes

A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a convex bounded polyhedron, for short called a polytope. Applications range from the very pure (number theory, toric Hilbert functions, Kostant’s partition function in representation theory) to the most applied (cryptography, integer programming, contingency tables). This paper is a survey of this problem and its applications. We review the basic structure theorems about this type of counting problem. Perhaps the most famous special case is the theory of Ehrhart polynomials, introduced in the 1960s by Eugène Ehrhart. These polynomials count the number of lattice points in the different integral dilations of an integral convex polytope. We discuss recent algorithmic solutions to this problem and conclude with a look at what happens when trying to count lattice points in more complicated regions of space.

All Linear and Integer Programs Are Slim 3-Way Transportation Programs

We show that any rational convex polytope is polynomial-time representable as a 3-way line-sum transportation polytope of “slim”

Vertices of Gelfand--Tsetlin Polytopes

(r,c,3)

An effective version of Pólya's theorem on positive definite forms

format. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. We provide a polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs. Our construction resolves several standing problems on

Graphs of transportation polytopes

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