Two disjoint 5-holes in point sets

Scheucher, Manfred

doi:10.1016/j.comgeo.2020.101670

Cited by 10 publications

(7 citation statements)

References 19 publications

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“…However, it might be possible to further optimize the SAT model to make the solver terminate faster (cf. [Sch20]) so that one obtains smaller certificates.…”

Section: Discussionmentioning

confidence: 99%

“…Kalbfleisch et al [KKS70]), and g (2) (6) = 17 was shown by Szekeres and Peters [SP06] using heavy computer assistance. While their computer program uses thousands of CPU hours, we have developed a SAT framework [Sch20] which allows to verify this result within only 2 CPU hours, and an independent verification of their result using SAT solvers was done by Marić [Mar19].…”

Section: Theorem 1 ([Es35]mentioning

confidence: 99%

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A SAT attack on higher dimensional Erdős--Szekeres numbers

Scheucher¹

2021

Preprint

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A famous result by Erdős and Szekeres (1935) asserts that, for every k, d ∈ N, there is a smallest integer n = g (d) (k), such that every set of at least n points in R d in general position contains a k-gon, i.e., a subset of k points which is in convex position. We present a SAT model for higher dimensional point sets which is based on chirotopes, and use modern SAT solvers to investigate Erdős-Szekeres numbers in dimensions d = 3, 4, 5. We show g (3) (7) ≤ 13, g (4) (8) ≤ 13, and g (5) (9) ≤ 13, which are the first improvements for decades. For the setting of k-holes (i.e., k-gons with no other points in the convex hull), where h (d) (k) denotes the minimum number n such that every set of at least n points in R d in general position contains a k-hole, we show h (3) (7) ≤ 14, h (4) (8) ≤ 13, and h (5) (9) ≤ 13. Moreover, all obtained bounds are sharp in the setting of chirotopes and we conjecture them to be sharp also in the original setting of point sets.

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“…However, it might be possible to further optimize the SAT model to make the solver terminate faster (cf. [Sch20]) so that one obtains smaller certificates.…”

Section: Discussionmentioning

confidence: 99%

Section: Theorem 1 ([Es35]mentioning

confidence: 99%

A SAT attack on higher dimensional Erdős--Szekeres numbers

Scheucher¹

2021

Preprint

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“…This can be quantified as follows. Let us conclude this section with an observation about interior-disjoint k-holes, as it emerged from a discussion with Manfred Scheucher (see also [34]). A k-hole of P is a subset of k points in convex position whose convex hull is disjoint from all other points in P. A k-hole and an -hole are called interior-disjoint, if their respective convex hulls are interior-disjoint (they can share up to two points).…”

Section: Unoriented Edges Lemmamentioning

confidence: 94%

Connectivity of Triangulation Flip Graphs in the Plane

Wagner

Welzl

2022

Discrete Comput Geom

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Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of P is a full triangulation of some subset $$P'$$ P ′ of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge (called edge flip), removes a non-extreme point of degree 3, or adds a point in $$P \setminus P'$$ P \ P ′ as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is $$\lceil {n}/{2}-2\rceil $$ ⌈ n / 2 - 2 ⌉ -vertex connected, and the bistellar flip graph is $$(n-3)$$ ( n - 3 ) -vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where $$(n-3)$$ ( n - 3 ) -vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension $$\lceil {n}/{2} -2\rceil $$ ⌈ n / 2 - 2 ⌉ (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension $$n-3$$ n - 3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance $$n-3$$ n - 3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations of a point set are regular iff the partial order of partial subdivisions has height $$n-3$$ n - 3 . (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations.

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“…The idea behind the SAT model is very similar as in [Sch20]: We assume towards a contradiction that cf(15) ≤ 3, that is, there is a set of 15 points with no 4-crossing family. We have Boolean variables X abc to indicate whether three points a, b, c are positively or negatively oriented.…”

Section: Sets Of 15 Points Always Contain a 4-crossing Familymentioning

confidence: 99%

“…We have Boolean variables X abc to indicate whether three points a, b, c are positively or negatively oriented. As outlined in [Sch20], these variables have to fulfill the signotope axioms [FW01,BFK15]. Based on the variables for triple orientations, we then assign auxiliary variables Y ab,cd to indicate whether the two segments ab, cd cross.…”

Section: Sets Of 15 Points Always Contain a 4-crossing Familymentioning

confidence: 99%

On Crossing-Families in Planar Point Sets

Aichholzer,

Kynčl,

Scheucher

et al. 2021

Preprint

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A k-crossing family in a point set S in general position is a set of k segments spanned by points of S such that all k segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4.(2) There are sets of n points which do not contain a crossing family of size larger than 8 n 41 . Both results improve the previously best known bounds.

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Two disjoint 5-holes in point sets

Cited by 10 publications

References 19 publications

A SAT attack on higher dimensional Erdős--Szekeres numbers

A SAT attack on higher dimensional Erdős--Szekeres numbers

Connectivity of Triangulation Flip Graphs in the Plane

On Crossing-Families in Planar Point Sets

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