2 Search Results for "Nielsen, Frank"


Document
Multimedia Contribution
On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)

Authors: Frank Nielsen and Laetitia Shao

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


Abstract
Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies.

Cite as

Frank Nielsen and Laetitia Shao. On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 67:1-67:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{nielsen_et_al:LIPIcs.SoCG.2017.67,
  author =	{Nielsen, Frank and Shao, Laetitia},
  title =	{{On Balls in a Hilbert Polygonal Geometry}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{67:1--67:4},
  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.2017.67},
  URN =		{urn:nbn:de:0030-drops-72443},
  doi =		{10.4230/LIPIcs.SoCG.2017.67},
  annote =	{Keywords: Projective geometry, Hilbert geometry, balls}
}
Document
How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking

Authors: Anna Adamaszek, Parinya Chalermsook, and Andreas Wiese

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)


Abstract
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axis-parallel rectangles in the plane. Our goal is to compute a maximum weight subset of pairwise non-overlapping rectangles. Due to its various applications, as well as connections to many other problems in computer science, MWISR has received a lot of attention from the computational geometry and the approximation algorithms community. However, despite being extensively studied, MWISR remains not very well understood in terms of polynomial time approximation algorithms, as there is a large gap between the upper and lower bounds, i.e., O(log n\ loglog n) v.s. NP-hardness. Another important, poorly understood question is whether one can color rectangles with at most O(omega(R)) colors where omega(R) is the size of a maximum clique in the intersection graph of a set of input rectangles R. Asplund and Grünbaum obtained an upper bound of O(omega(R)^2) about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR. In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of 1-delta for an arbitrarily small constant delta > 0. Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with O(omega(R)) colors which implies a constant upper bound on the integrality gap of the canonical LP. For some applications of MWISR the possibility to shrink the rectangles has a natural, well-motivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.

Cite as

Anna Adamaszek, Parinya Chalermsook, and Andreas Wiese. How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 43-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{adamaszek_et_al:LIPIcs.APPROX-RANDOM.2015.43,
  author =	{Adamaszek, Anna and Chalermsook, Parinya and Wiese, Andreas},
  title =	{{How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{43--60},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.43},
  URN =		{urn:nbn:de:0030-drops-52936},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.43},
  annote =	{Keywords: Approximation algorithms, independent set, resource augmentation, rectangle intersection graphs, PTAS}
}
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