5 Search Results for "Keszegh, Balázs"


Document
An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

Authors: Eyal Ackerman, Balázs Keszegh, and Günter Rote

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


Abstract
What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ⌈ n/6 ⌉), for m ≥ n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C.

Cite as

Eyal Ackerman, Balázs Keszegh, and Günter Rote. An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2020.1,
  author =	{Ackerman, Eyal and Keszegh, Bal\'{a}zs and Rote, G\"{u}nter},
  title =	{{An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{1:1--1:18},
  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.2020.1},
  URN =		{urn:nbn:de:0030-drops-121591},
  doi =		{10.4230/LIPIcs.SoCG.2020.1},
  annote =	{Keywords: Simple polygon, Ramsey theory, combinatorial geometry}
}
Document
Radon Numbers Grow Linearly

Authors: Dömötör Pálvölgyi

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


Abstract
Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_k ≤ c(r₂)⋅ k.

Cite as

Dömötör Pálvölgyi. Radon Numbers Grow Linearly. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 60:1-60:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{palvolgyi:LIPIcs.SoCG.2020.60,
  author =	{P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Radon Numbers Grow Linearly}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{60:1--60:5},
  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.2020.60},
  URN =		{urn:nbn:de:0030-drops-122183},
  doi =		{10.4230/LIPIcs.SoCG.2020.60},
  annote =	{Keywords: discrete geometry, convexity space, Radon number}
}
Document
Coloring Intersection Hypergraphs of Pseudo-Disks

Authors: Balázs Keszegh

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


Abstract
We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.

Cite as

Balázs Keszegh. Coloring Intersection Hypergraphs of Pseudo-Disks. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{keszegh:LIPIcs.SoCG.2018.52,
  author =	{Keszegh, Bal\'{a}zs},
  title =	{{Coloring Intersection Hypergraphs of Pseudo-Disks}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{52:1--52:15},
  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.2018.52},
  URN =		{urn:nbn:de:0030-drops-87657},
  doi =		{10.4230/LIPIcs.SoCG.2018.52},
  annote =	{Keywords: combinatorial geometry, conflict-free coloring, geometric hypergraph coloring}
}
Document
Proper Coloring of Geometric Hypergraphs

Authors: Balázs Keszegh and Dömötör Pálvölgyi

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)


Abstract
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.

Cite as

Balázs Keszegh and Dömötör Pálvölgyi. Proper Coloring of Geometric Hypergraphs. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{keszegh_et_al:LIPIcs.SoCG.2017.47,
  author =	{Keszegh, Bal\'{a}zs and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{Proper Coloring of Geometric Hypergraphs}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{47:1--47:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.47},
  URN =		{urn:nbn:de:0030-drops-71926},
  doi =		{10.4230/LIPIcs.SoCG.2017.47},
  annote =	{Keywords: discrete geometry, decomposition of multiple coverings, geometric hypergraph coloring}
}
Document
Coloring Points with Respect to Squares

Authors: Eyal Ackerman, Balázs Keszegh, and Máté Vizer

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)


Abstract
We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

Cite as

Eyal Ackerman, Balázs Keszegh, and Máté Vizer. Coloring Points with Respect to Squares. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2016.5,
  author =	{Ackerman, Eyal and Keszegh, Bal\'{a}zs and Vizer, M\'{a}t\'{e}},
  title =	{{Coloring Points with Respect to Squares}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{5:1--5:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.5},
  URN =		{urn:nbn:de:0030-drops-58972},
  doi =		{10.4230/LIPIcs.SoCG.2016.5},
  annote =	{Keywords: Geometric hypergraph coloring, Polychromatic coloring, Homothets, Cover-decomposability}
}
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