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- Jesús A. De Loera, Raymond Hemmecke, Jeremiah Tauzer, Ruriko Yoshida
- J. Symb. Comput.
- 2004

- Jesús A. De Loera, David Haws, Raymond Hemmecke, Peter Huggins, Bernd Sturmfels, Ruriko Yoshida
- J. Symb. Comput.
- 2004

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 Barvinok (1994); Barvinok and Woods (2003). Under the assumption that d, n are fixed, this representation allows us to compute the Graver basis and the reduced Gröbner basis of the ideal I A ,… (More)

- Jesús A. De Loera, Raymond Hemmecke, Shmuel Onn, Robert Weismantel
- Discrete Optimization
- 2008

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 so-called directed augmentation, and the stabilization of certain… (More)

- Jesús A. De Loera, Raymond Hemmecke, Matthias Köppe, Robert Weismantel
- Math. Oper. Res.
- 2006

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds… (More)

- Jesús A. De Loera, Raymond Hemmecke, Matthias Köppe
- INFORMS Journal on Computing
- 2009

We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a… (More)

Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes. Magic cubes and squares are very popular combinatorial objects (see [2, 15, 17] and their references). A magic square is a square matrix whose entries are nonnegative… (More)

- Raymond Hemmecke
- 2003

In this paper we extend test set based augmentation methods for integer linear programs to programs with more general convex objective functions. We show existence and computability of finite test sets for these wider problem classes by providing an explicit relationship to Graver bases. One candidate where this new approach may turn out fruitful is the… (More)

- Raymond Hemmecke, Robert Weismantel
- SIAM Journal on Optimization
- 2007

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 (1994); Barvinok and Woods (2003). Under the assumption that d and n are fixed, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal I A… (More)

- Raymond Hemmecke, Matthias Köppe, Jon Lee, Robert Weismantel
- 50 Years of Integer Programming
- 2010

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of… (More)