Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

Schewe, Lars

doi:10.1007/s00454-009-9222-y

Cited by 20 publications

(25 citation statements)

References 28 publications

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“…This section is devoted to defining oriented matroids and explaining how the nonexistence of certain oriented matroids (and consequently the non-existence of certain point sets) can be shown using SAT solvers. The use of SAT solvers to generate oriented matroids was first described in [11] and [12]. The method used there is the basis for the approach outlined in this section.…”

Section: Generation Of Oriented Matroids Using Sat Solversmentioning

confidence: 99%

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Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

Bremner

Schewe

2011

Experimental Mathematics

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Abstract. We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d = 6. This implies that for all pairs (d, n) with n−d ≤ 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n − d ≤ 6. We show this result by showing this bound for a more general structure -so-called matroid polytopes -by reduction to a small number of satisfiability problems. We show that the d-step conjecture is true in dimension 6 as well. We derive this result by considering a more general class of objects, namely matroid polytopes, i.e. oriented matroids, which, if realizable, correspond to convex polytopes. We show that no 6-dimensional matroid polytope with 12 vertices and a facet path of length 7 exists. Then ∆(6, 12) = 6 follows by considering polarity and the already known bounds.To show that ∆(6, 12) ≤ 6 we first give combinatorial conditions for matroid polytopes that violate this bound. This achieved through the study of path complexes (see Bremner et al. [2], cf. Section 1). We then show that these conditions cannot be satisfied by an oriented matroid. To show this we use a satisfiability solver to produce the desired contradiction (see Section 2). We will use the same method to show that ∆(4, 11) = 6, which settles another special case of the Hirsch conjecture. The latter result allows us to also improve the upper bound on ∆(5, 12) from 9 to 8.For small parameters there are already known general bounds that allow us to compute or at least bound the diameter of polytopes. We summarize them in the following Lemma. For an overview about the known bounds we refer to the respective chapters in the books by Grünbaum [6] (1) ∆(3, n) =

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Section: Generation Of Oriented Matroids Using Sat Solversmentioning

confidence: 99%

“…For details we refer to [11,12]. It remains to explain how to encode the facet path and the forbidden shortcuts.…”

Section: Generation Of Oriented Matroids Using Sat Solversmentioning

confidence: 99%

Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

Bremner

Schewe

2011

Experimental Mathematics

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“…The alternating property implies that we only need a variable for the natural ordering of a given set. Schewe [12] shows how to define clauses of a conjunctive normal form statement that are necessary to satisfy the second condition of Definition 2.1. For each r-element subset σ of [n+ 2] and each 4-element subset of [n+ 2]\σ, there are 16 clauses with 6 literals each that must be satisfied.…”

Section: Oriented Matroid Programmingmentioning

confidence: 99%

A Proof of the Strict Monotone 5-Step Conjecture

Gallagher

Morris

2019

Discrete Comput Geom

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A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus ∆sm(5, 10) = 5. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that ∆sm(4, 9) = 5 and ∆m(4, 9) = ∆m(5, 10) = 6. We shorten the known non-computer proof of the strict monotone 4-step conjecture.

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“…However it is possible, as explained in [14], to reduce this axiomatization to a (relatively) short list of axioms with a clear combinatorial or geometrical interpretation of each axiom in a maner very similar to the known simple axiomatizations of pseudoline arrangements [3,15]. This open the door to the generation of double pseudoline arrangements with prescribed properties using, for example, satisfiability solvers as proposed in [16,17], see also [5,7,2,8,6]. In particular it will be interesting to generate the arrangements that maximize the number of connected components of the intersection of the Möbius strips surrounded by the double pseudolines.…”

Section: Further Developmentsmentioning

confidence: 99%