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Triangulations: Structures for Algorithms and Applications
Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents…
Effective lattice point counting in rational convex polytopes
How to integrate a polynomial over a simplex
- V. Baldoni, Nicole Berline, J. D. Loera, M. Köppe, M. Vergne
- Mathematics, Computer ScienceMath. Comput.
- 11 September 2008
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.
Gröbner bases and triangulations of the second hypersimplex
A quadratic Gröbner basis is presented for the associated toric idealK(Kn) and a non-regular triangulation of Δ(2,n) forn≥9 is constructed.
A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility
We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a…
A Polytopal Generalization of Sperner's Lemma
This work provides two proofs of the following conjecture: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument.
The Polytope of All Triangulations of a Point Configuration
We study the convex hull P A of the 0-1 incidence vectors of all triangulations of a point con guration A. This was called the universal polytope in . The a ne span of P A is described in terms of…
The many aspects of counting lattice points in polytopes
- J. D. Loera
- 1 August 2005
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…
Integer Polynomial Optimization in Fixed Dimension
- J. D. Loera, R. Hemmecke, M. Köppe, R. Weismantel
- Mathematics, Computer ScienceMath. Oper. Res.
- 5 October 2004
For the optimization of an integer polynomial over the lattice points of a convex polytope, an algorithm is shown to compute lower and upper bounds for the optimal value.