Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm
Matthias Köppe,Sven Verdoolaege +1 more
TL;DR: 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.
read more
Abstract: Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion–exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using partially open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Nonlinear Integer Programming
Raymond Hemmecke,Matthias Köppe,Jon Lee,Robert Weismantel +3 more
- 01 Dec 2010
TL;DR: This chapter is a study of a simple version of general nonlinear integer problems, where all constraints are still linear, and focuses on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure.
323
Computer Algebra and Polynomials
Jaime Gutierrez,Josef Schicho,Martin Weimann +2 more
- 01 Jan 2015
TL;DR: The theory of integer points in polyhedra has been studied extensively in enumerative combinatorics, e.g. in this paper, where Ehrhart's method for proving that a counting function is a polynomial is discussed.
59
•Posted Content
The power of pyramid decomposition in Normaliz
TL;DR: The use of pyramid decomposition in Normaliz is described, a software tool for the computation of Hilbert bases and enumerative data of rational cones and affine monoids, to process triangulations of size that arise in the computations of Ehrhart series related to the theory of social choice.
49
Counting with rational generating functions
Sven Verdoolaege,Kevin Woods +1 more
TL;DR: This paper proves that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations.
44
h*-POLYNOMIALS OF ZONOTOPES
TL;DR: The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P as discussed by the authors.
43
References
•Book
Discriminants, Resultants, and Multidimensional Determinants
Izrailʹ Moiseevich Gelʹfand,Mikhail Kapranov,Andrei Zelevinsky +2 more
- 10 May 2013
TL;DR: The Cayley method of studying discriminants was used by Cayley as discussed by the authors to study the Cayley Method of Discriminants and Resultants for Polynomials in One Variable and for forms in Several Variables.
3.1K
Kneser's conjecture, chromatic number, and homotopy
TL;DR: If the simplicial complex formed by the neighborhoods of points of a graph is (k − 2)-connected then the graph is not k-colorable, and Kneser's conjecture is proved, asserting that if all n-subsets of a (2n − k)-element set are divided into k + 1 classes, one of the classes contains two disjoint n- subsets.
1K
Effective lattice point counting in rational convex polytopes
TL;DR: LattE , a computer package for lattice point enumeration which contains the first implementation of A. Barvinok’s algorithm, is described and it is proved that these kinds of symbolic–algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting.
364
•Book
Using the Borsuk-Ulam theorem : lectures on topological methods in combinatorics and geometry
Jiří Matoušek,Anders Björner,Günter M. Ziegler +2 more
- 01 Jan 2008
TL;DR: The Borsuk-Ulam Theorem and its application in topological interludes can be found in this paper, where maps and non-embeddability are discussed.
364
Points entiers dans les polyèdres convexes
TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).