Many compiler optimization techniques depend on the ability to calculate the number of
elements that satisfy certain conditions. If these conditions can be represented by linear …

It is undeniable that geometric ideas have been very important to the foundations of modern
discrete optimization. The influence that geometric algorithms have in optimization was …

Revised November 2006; Accepted January 2007 bstract The theory of duality for linear
programs is well-developed and has been successful in advancing both the theory and …

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 …

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The
new algorithms are based on irrational signed decomposition in the primal space and the …

We define a violator operator which captures the definition of a minimal Gr\" obner basis of
an ideal. This construction places the problem of computing a Gr\" obner basis within the …

We classify, according to their computational complexity, integer optimization problems
whose constraints and objective functions are polynomials with integer coefficients, and the …

We explore the computational complexity of computing pure Nash equilibria for a new class
of strategic games called integer programming games, with differences of piecewise-linear …

We investigate properties of Ehrhart polynomials for matroid polytopes, independence
matroid polytopes, and polymatroids. In the first half of the paper we prove that, for fixed …

Presburger arithmetic is the first-order theory of the natural numbers with addition (but no
multiplication). We characterize sets that can be defined by a Presburger formula as exactly …