J. A. De Loera

Short rational functions for toric algebra and applications

Abstract We encode the binomials belonging to the toric ideal I A associated with an integral d × n matrix A using a short sum of rational functions as introduced by Barvinok (Math. Operations Research 19 (1994) 769) and Barvinok and Woods (J. Amer. Math. Soc. 16 (2003) 957). Under the assumption that d and n are fixed, this representation allows us to compute a universal Grobner basis and the reduced Grobner basis of the ideal I A , with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate applications to enumerative combinatorics, integer programming, and statistics.

Counting Integer Flows in Networks

Abstract This paper discusses analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory, representation theory, and statistics. Our methods are based on the study of rational functions with poles on arrangements of hyperplanes; they surpass traditional exhaustive enumeration and can even yield formulas when the input data contains some parameters. We also discuss the calculation of chambers in detail because it is a necessary subroutine.

Software for exact integration of polynomials over polyhedra

We are interested in quickly computing the exact value of integrals of polynomial functions over domains that are decomposable into convex polyhedra (e.g., a tetrahedral or cubical mesh decomposition of space). We describe a software implementation, part of the software LattE, and provide benchmark computations.

A generating function for all semi-magic squares and the volume of the Birkhoff polytope

We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of Bn and any of its faces.

Hilbert's nullstellensatz and an algorithm for proving combinatorial infeasibility

Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilberts Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

MINIMAL SIMPLICIAL DISSECTIONS AND TRIANGULATIONS OF CONVEX 3-POLYTOPES.

Abstract. This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P .• Can the minimal number of tetrahedra in a triangulation be decreased if one allows the use of interior points of P as vertices? • Can a dissection of P use fewer tetrahedra than a triangulation? • Does the size of a minimal triangulation depend on the geometric realization of P ? The main result of this paper is that all these questions have an affirmative answer. Even stronger, the gaps of size produced by allowing interior vertices or by using dissections may be linear in the number of points.

Three Kinds of Integer Programming Algorithms Based on Barvinok’s Rational Functions

This paper presents three kinds of algebraic-analytic algorithms for solving integer and mixed integer programming problems. We report both theoretical and experimental results. We use the generating function techniques introduced by A. Barvinok.

A computational study of integer programming algorithms based on Barvinok's rational functions

This paper discusses five algorithms to solve linear integer programming problems that use the rational function techniques introduced by A. Barvinok. We report on the first ever experimental results based on these techniques.

The Hierarchy of Circuit Diameters and Transportation Polytopes

Abstract The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still unknown whether the Hirsch conjecture is true for general m × n -transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. This hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarchy of diameters for the m × n -transportation polytopes. We show that the Hirsch conjecture bound of m + n − 1 is actually valid in most of these diameter notions. Second, we prove that for 3 × n -transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2 × n -transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter.

FPTAS for mixed-integer polynomial optimization with a fixed number of variables

We show the existence of an FPTAS for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.