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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)
The motivation for the paper is the geometric approach to learning Bayesian network (BN) structure. The basic idea of our approach is to represent every BN structure by a certain uniquely determined vector so that usual scores for learning BN structure become affine functions of the vector representative. The original proposal from Studený et al. (2010)(More)
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)
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)
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)
An integer feasibility problem is a fundamental problem in many areas, such as operations research, number theory, and statistics. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite set of vectors in Z d and its saturation. In this paper we present an algorithm to compute an explicit(More)