Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

  title={Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems},
  author={Felix Breuer and Zafeirakis Zafeirakopoulos},
  journal={Annals of Combinatorics},
Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omegacombines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as… 

Matrix computations with the Omega calculus

An extension of the Omega calculus in the context of matrix analysis introduced recently by Neto is explored, obtaining Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular.

Computing efficiently the non-properness set of polynomial maps on the plane

The algorithm takes into account the sparsity of polynomials as it depends on (the Minkowski sum of) the Newton polytopes of f and provides a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets that correspond to edges of the Newtonpolytope, which is of independent interest.

A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied

Partition analysis, modular functions, and computer algebra

This article describes recent developments connecting problems of enumerative combinatorics, constrained by linear systems of Diophantine inequalities, with number theory topics like partitions,

Generating Functions of Weighted Voting Games, MacMahon's Partition Analysis, and Clifford Algebras

  • A. Neto
  • Mathematics
    Math. Oper. Res.
  • 2019
A new generalized generating function is introduced that gives new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not.

Interview with Peter Paule

Peter Paule obtained a Ph.D. from the University of Vienna in 1982 under the supervision of Johann Cigler. In 1996 he earned a habilitation from Johannes Kepler University. Since 1990 he has held a

Matrix Analysis and Omega Calculus

  • A. Neto
  • Computer Science, Mathematics
    SIAM Rev.
  • 2020
A new operator based approach to matrix analysis is introduced that is an extension of a tool introduced long ago by MacMahon to analyze the partitions of partitions in matrix analysis.

Plane partitions in the work of Richard Stanley and his school

These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics



A Euclid style algorithm for MacMahon's partition analysis

  • G. Xin
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2015

Effective lattice point counting in rational convex polytopes

MacMahon’s Partition Analysis V: Bijections, Recursions, and Magic Squares

A significant portion of MacMahon’s famous book “Combinatory Analysis” is devoted to the development of “Partition Analysis” as a computational method for solving problems in connection with linear

MacMahon's Partition Analysis VI: A New Reduction Algorithm

A significant algorithmic improvement of the Omega package is presented, which overcomes a problem related to the computational treatment of roots of unity and turns out to be superior to "The Method of Elliott" which is described in MacMahon's book.

Theory of linear and integer programming

  • A. Schrijver
  • Mathematics
    Wiley-Interscience series in discrete mathematics and optimization
  • 1999
Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear

Lectures on Polytopes

Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward

Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm

It is proved that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects, and all linear identities in the space of indicator function identities can be purely expressed using partially open variants of the full-dimensionalpolyhedra in the identity.

On Counting Integral Points in a Convex Rational Polytope

Two variants of an inversion algorithm are provided that provide the Ehrhart polynomial of a convex integer polytope and an alternative that avoids the complex integration of F( z) and whose main computational effort is to solve a linear system.

Short rational generating functions for lattice point problems

Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we

How to integrate a polynomial over a simplex

It is proved that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus, and if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, it is proven that integration can be done inPolynomial time.