3 Search Results for "Horton, Michael"

Document
DOI: 10.4230/LIPIcs.ITP.2024.35
Formal Verification of the Empty Hexagon Number

Authors: Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position (i.e., no three points on a common line) contains an empty convex hexagon. Heule and Scheucher solved this problem with a combination of geometric insights and automated reasoning techniques by constructing CNF formulas ϕ_n, with O(n⁴) clauses, such that if ϕ_n is unsatisfiable then every set of n points in general position must contain an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17 300 CPU hours of parallel computation. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers ideas in discrete computational geometry and SAT encoding techniques by introducing a framework that connects geometric objects to propositional assignments. We see this as a key step towards the formal verification of other SAT-based results in geometry, since the abstractions we use have been successfully applied to similar problems. Overall, we hope that our work sets a new standard for the verification of geometry problems relying on extensive computation, and that it increases the trust the mathematical community places in computer-assisted proofs.
Cite as

Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule. Formal Verification of the Empty Hexagon Number. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{subercaseaux_et_al:LIPIcs.ITP.2024.35,
  author =	{Subercaseaux, Bernardo and Nawrocki, Wojciech and Gallicchio, James and Codel, Cayden and Carneiro, Mario and Heule, Marijn J. H.},
  title =	{{Formal Verification of the Empty Hexagon Number}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{35:1--35:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.35},
  URN =		{urn:nbn:de:0030-drops-207633},
  doi =		{10.4230/LIPIcs.ITP.2024.35},
  annote =	{Keywords: Empty Hexagon Number, Discrete Computational Geometry, Erd\H{o}s-Szekeres}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.10
Erdős-Szekeres-Type Problems in the Real Projective Plane

Authors: Martin Balko, Manfred Scheucher, and Pavel Valtr

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We consider point sets in the real projective plane ℝ𝒫² and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős-Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős-Szekeres theorem about point sets in convex position in ℝ𝒫², which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝ𝒫² agrees with the definition of convex sets introduced by Steinitz in 1913. For k ≥ 3, an (affine) k-hole in a finite set S ⊆ ℝ² is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from ℝ𝒫², called projective k-holes, we find arbitrarily large finite sets of points from ℝ𝒫² with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for k ≤ 7. On the other hand, we show that the number of k-holes can be substantially larger in ℝ𝒫² than in ℝ² by constructing, for every k ∈ {3,… ,6}, sets of n points from ℝ² ⊂ ℝ𝒫² with Ω(n^{3-3/5k}) projective k-holes and only O(n²) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝ𝒫² and about some algorithmic aspects. The study of extremal problems about point sets in ℝ𝒫² opens a new area of research, which we support by posing several open problems.
Cite as

Martin Balko, Manfred Scheucher, and Pavel Valtr. Erdős-Szekeres-Type Problems in the Real Projective Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balko_et_al:LIPIcs.SoCG.2022.10,
  author =	{Balko, Martin and Scheucher, Manfred and Valtr, Pavel},
  title =	{{Erd\H{o}s-Szekeres-Type Problems in the Real Projective Plane}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.10},
  URN =		{urn:nbn:de:0030-drops-160182},
  doi =		{10.4230/LIPIcs.SoCG.2022.10},
  annote =	{Keywords: real projective plane, point set, convex position, k-gon, k-hole, Erd\H{o}s-Szekeres theorem, Horton set, random point set}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.7
On β-Plurality Points in Spatial Voting Games

Authors: Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Let V be a set of n points in ℝ^d, called voters. A point p ∈ ℝ^d is a plurality point for V when the following holds: for every q ∈ ℝ^d the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0<β⩽1. We investigate the existence and computation of β-plurality points, and obtain the following results. - Define β^*_d := sup{β : any finite multiset V in ℝ^d admits a β-plurality point}. We prove that β^*₂ = √3/2, and that 1/√d ⩽ β^*_d ⩽ √3/2 for all d⩾3. - Define β(V) := sup {β : V admits a β-plurality point}. We present an algorithm that, given a voter set V in {ℝ}^d, computes an (1-ε)⋅ β(V) plurality point in time O(n²/ε^(3d-2) ⋅ log(n/ε^(d-1)) ⋅ log²(1/ε)).
Cite as

Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton. On β-Plurality Points in Spatial Voting Games. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{aronov_et_al:LIPIcs.SoCG.2020.7,
  author =	{Aronov, Boris and de Berg, Mark and Gudmundsson, Joachim and Horton, Michael},
  title =	{{On \beta-Plurality Points in Spatial Voting Games}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.7},
  URN =		{urn:nbn:de:0030-drops-121651},
  doi =		{10.4230/LIPIcs.SoCG.2020.7},
  annote =	{Keywords: Computational geometry, Spatial voting theory, Plurality point, Computational social choice}
}
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