Triangulations

Loera, Jesús A. De; Rambau, Jörg; Santos, Francisco

doi:10.1007/978-3-642-12971-1

Cited by 305 publications

(184 citation statements)

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“…This raises the issue of finding the smallest triangulation possible of a given polytope. Unfortunately this problem was proved to be NP-hard even for fixed dimension three (see [15]). Thus it is in general not a good idea to spend time finding the smallest triangulation possible.…”

Section: Triangulation Of Arbitrary Polytopesmentioning

confidence: 99%

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How to integrate a polynomial over a simplex

Berline²,

et al. 2010

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Abstract. This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes and discussion of other available methods.

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Section: Triangulation Of Arbitrary Polytopesmentioning

confidence: 99%

“…A cautionary remark is that one can naively assume that triangulations help for nonconvex polyhedral regions, while in reality it does not because there exist nonconvex polyhedra that are not triangulable unless one adds new points. Deciding how many new points are necessary is an NPhard problem [15].…”

Section: Triangulation Of Arbitrary Polytopesmentioning

confidence: 99%

How to integrate a polynomial over a simplex

Berline²,

et al. 2010

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“…with t given in (7). A simple calculation shows that 0 ≤ t < 1/2 and 0 ≤ c − 1/2 + t ≤ 5/2 for 0 ≤ c ≤ 3.…”

Section: Proof the Exact Value Of St(s) Ismentioning

confidence: 98%

“…Triangulations are an important geometric construct which is used in many algorithmic applications, and are also an interesting object of study in discrete and combinatorial geometry (recent comprehensive surveys can be found in [7,17]). …”

Section: Introductionmentioning

confidence: 99%

Counting Plane Graphs: Flippability and Its Applications

Hoffmann

Sharir

Sheffer

et al. 2011

Lecture Notes in Computer Science

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We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S.We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N ) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N ) < 30 N ) that any N -element point set admits at most 6. forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN , fewer than cN , or more than cN edges, for any constant parameter c, in terms of c and N .

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“…rather than 2 n n+1 . The factor n!, the maximum number of simplices in a triangulation of the n-cube [36], is the lowest order that guarantees deriving the convex hull of Equation (22) on the n-cube.…”

Section: Exploiting Dominance Relationsmentioning

confidence: 99%

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

Misener

Smadbeck

Floudas

2014

Optimization Methods and Software

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The Global Mixed-Integer Quadratic Optimizer, GloMIQO, addresses mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation-linearisation technique equations, convex multivariable terms, αBB convexifications, and low-and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [35] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression; algorithmic descriptions for deriving each class of cut; presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.

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Triangulations

Cited by 305 publications

References 0 publications

How to integrate a polynomial over a simplex

How to integrate a polynomial over a simplex

Counting Plane Graphs: Flippability and Its Applications

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

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