Topological Prismatoids and Small Simplicial Spheres of Large Diameter

Criado, Francisco; Santos, Francisco

doi:10.1080/10586458.2019.1641766

Cited by 10 publications

(18 citation statements)

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“…This bound contrasts the best-known upper bounds on polytope diameters, which are linear in fixed dimension, but grow exponentially in the dimension (e.g., [18] and [10]). For a survey of the best bounds and more updates about diameters of polytopes see [7,8,10,26,28] and the references therein.…”

Section: Our Resultsmentioning

confidence: 99%

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Adler¹,

Loera²,

Klee³

et al. 2021

Electron. J. Combin.

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Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.

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Section: Our Resultsmentioning

confidence: 99%

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Adler¹,

Loera²,

Klee³

et al. 2021

Electron. J. Combin.

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“…Finally in Section 5, we discuss the implementation of our algorithm, describe the relation of our certificates to classical final polynomials, and list the results of our computations regarding the realizability of a database of selected spheres. In particular, we derive non-realizability certificates for a large number of new instances of prismatoids, a class of polytopes introduced in [6], recover the recent result of [24] on the non-realizability of Jockusch's family of simplicial 3-spheres (see [23]), as well as providing a few other examples of new or simpler non-realizability certificates.…”

Section: Introductionmentioning

confidence: 88%

“…In the beginning of the section we established our general approach, which is essentially that of solving an instance of problem (6). In this subsection we discuss additional details of the actual instantiation, explaining our implementation.…”

Section: Implementation Detailsmentioning

confidence: 99%

“…Example 16. Let P be prismatoid #3513 from [6]. It is 5-dimensional and has 14 vertices and 94 facets, two of which are non-simplicial.…”

Section: Vertex Fixingmentioning

confidence: 99%

“…In [6] Criado and Santos constructed 4093 abstract 5-dimensional non-d-step prismatoids with number of vertices between 14 and 28. These are examples of non-Hirsch spheres but it is not known if any of them is realizable as a convex polytope.…”

Section: Prismatoidsmentioning

confidence: 99%

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General non-realizability certificates for spheres with linear programming

Gouveia¹,

Macchia²,

Wiebe³

2021

Preprint

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In this paper we present a simple technique to derive certificates of non-realizability for an abstract polytopal sphere. Our approach uses a variant of the classical algebraic certificates introduced by Bokowski and Sturmfels in [5], the final polynomials. More specifically we reduce the problem of finding a realization to that of finding a positive point in a variety and try to find a polynomial with positive coefficients in the generating ideal (a positive polynomial), showing that such point does not exist. Many, if not most, of the techniques for proving non-realizability developed in the last three decades can be seen as following this framework, using more or less elaborate ways of constructing such positive polynomials. Our proposal is more straightforward as we simply use linear programming to exhaustively search for such positive polynomials in the ideal restricted to some linear subspace. Somewhat surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives, and allows us to derive new examples of non-realizable abstract polytopal spheres.

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