Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra

@article{Baldoni2012ComputationOT,
  title={Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra},
  author={Velleda Baldoni and Nicole Berline and Jes{\'u}s A. De Loera and Matthias K{\"o}ppe and Mich{\`e}le Vergne},
  journal={Foundations of Computational Mathematics},
  year={2012},
  volume={12},
  pages={435-469}
}
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok in the… 
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References

SHOWING 1-10 OF 43 REFERENCES
Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory
TLDR
An algorithm is provided to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials, which are naturally valid for real (not just integer) dilations and thus provide a direct approach to real EHRhart theory.
Computing the Ehrhart quasi-polynomial of a rational simplex
TLDR
A polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex for rational polytopes of a fixed dimension is presented.
The many aspects of counting lattice points in polytopes
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. Applications
Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra
AbstractLet σ be a simplex of RN with vertices in the integral lattice ZN . The number of lattice points of mσ(={mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0 . In this paper we present: (i)
Residue formulae, vector partition functions and lattice points in rational polytopes
We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an
Lattice points in simple polytopes
in terms of fP(h) q(x)dx where the polytope P(h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of
Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra
Preface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville
How to integrate a polynomial over a simplex
TLDR
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.
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