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Documents authored by Patáková, Zuzana

Document
DOI: 10.4230/LIPIcs.SoCG.2022.57
Covering Points by Hyperplanes and Related Problems

Authors: Zuzana Patáková and Micha Sharir

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

Abstract
For a set P of n points in ℝ^d, for any d ≥ 2, a hyperplane h is called k-rich with respect to P if it contains at least k points of P. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of k-rich hyperplanes in ℝ^d, d ≥ 3, is at least Ω(n^d/k^α + n/k), with a sufficiently large constant of proportionality and with d ≤ α < 2d-1, then there exists a (d-2)-flat that contains Ω(k^{(2d-1-α)/(d-1)}) points of P. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for k-rich spheres.
Cite as

Zuzana Patáková and Micha Sharir. Covering Points by Hyperplanes and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 57:1-57:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{patakova_et_al:LIPIcs.SoCG.2022.57,
  author =	{Pat\'{a}kov\'{a}, Zuzana and Sharir, Micha},
  title =	{{Covering Points by Hyperplanes and Related Problems}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{57:1--57:7},
  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.2022.57},
  URN =		{urn:nbn:de:0030-drops-160652},
  doi =		{10.4230/LIPIcs.SoCG.2022.57},
  annote =	{Keywords: Rich hyperplanes, Incidences, Covering points by hyperplanes}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.40
A Stepping-Up Lemma for Topological Set Systems

Authors: Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d+1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.
Cite as

Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková. A Stepping-Up Lemma for Topological Set Systems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2021.40,
  author =	{Goaoc, Xavier and Holmsen, Andreas F. and Pat\'{a}kov\'{a}, Zuzana},
  title =	{{A Stepping-Up Lemma for Topological Set Systems}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.40},
  URN =		{urn:nbn:de:0030-drops-138396},
  doi =		{10.4230/LIPIcs.SoCG.2021.40},
  annote =	{Keywords: Helly-type theorem, Topological combinatorics, Homological minors, Stair convexity, Cubical complexes, Homological VC dimension, Ramsey-type theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.61
Bounding Radon Number via Betti Numbers

Authors: Zuzana Patáková

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

Abstract
We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.
Cite as

Zuzana Patáková. Bounding Radon Number via Betti Numbers. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{patakova:LIPIcs.SoCG.2020.61,
  author =	{Pat\'{a}kov\'{a}, Zuzana},
  title =	{{Bounding Radon Number via Betti Numbers}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{61:1--61:13},
  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.2020.61},
  URN =		{urn:nbn:de:0030-drops-122198},
  doi =		{10.4230/LIPIcs.SoCG.2020.61},
  annote =	{Keywords: Radon number, topological complexity, constrained chain maps, fractional Helly theorem, convexity spaces}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.62
Barycentric Cuts Through a Convex Body

Authors: Zuzana Patáková, Martin Tancer, and Uli Wagner

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

Abstract
Let K be a convex body in ℝⁿ (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K ∩ h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p₀ is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.
Cite as

Zuzana Patáková, Martin Tancer, and Uli Wagner. Barycentric Cuts Through a Convex Body. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{patakova_et_al:LIPIcs.SoCG.2020.62,
  author =	{Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Barycentric Cuts Through a Convex Body}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{62:1--62:16},
  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.2020.62},
  URN =		{urn:nbn:de:0030-drops-122201},
  doi =		{10.4230/LIPIcs.SoCG.2020.62},
  annote =	{Keywords: convex body, barycenter, Tukey depth, smooth manifold, critical points}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.41
Shellability is NP-Complete

Authors: Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We prove that for every d >= 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d >= 2 and k >= 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d >= 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.
Cite as

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. Shellability is NP-Complete. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2018.41,
  author =	{Goaoc, Xavier and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Shellability is NP-Complete}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{41:1--41:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.41},
  URN =		{urn:nbn:de:0030-drops-87542},
  doi =		{10.4230/LIPIcs.SoCG.2018.41},
  annote =	{Keywords: Shellability, simplicial complexes, NP-completeness, collapsibility}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.476
On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result

Authors: Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2. Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
Cite as

Xavier Goaoc, Isaac Mabillard, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 476-490, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goaoc_et_al:LIPIcs.SOCG.2015.476,
  author =	{Goaoc, Xavier and Mabillard, Isaac and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{476--490},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.476},
  URN =		{urn:nbn:de:0030-drops-51256},
  doi =		{10.4230/LIPIcs.SOCG.2015.476},
  annote =	{Keywords: Heawood Inequality, Embeddings, Van Kampen–Flores, Manifolds}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.507
Bounding Helly Numbers via Betti Numbers

Authors: Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.
Cite as

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner. Bounding Helly Numbers via Betti Numbers. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 507-521, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goaoc_et_al:LIPIcs.SOCG.2015.507,
  author =	{Goaoc, Xavier and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Bounding Helly Numbers via Betti Numbers}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{507--521},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.507},
  URN =		{urn:nbn:de:0030-drops-51297},
  doi =		{10.4230/LIPIcs.SOCG.2015.507},
  annote =	{Keywords: Helly-type theorem, Ramsey’s theorem, Embedding of simplicial complexes, Homological almost-embedding, Betti numbers}
}
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