4 Search Results for "Tardos, Gábor"

Document
DOI: 10.4230/LIPIcs.SoCG.2022.56
Disjointness Graphs of Short Polygonal Chains

Authors: János Pach, Gábor Tardos, and Géza Tóth

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

Abstract
The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G). Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G))³+ω(G).
Cite as

János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Short Polygonal Chains. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2022.56,
  author =	{Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za},
  title =	{{Disjointness Graphs of Short Polygonal Chains}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{56:1--56:12},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.56},
  URN =		{urn:nbn:de:0030-drops-160645},
  doi =		{10.4230/LIPIcs.SoCG.2022.56},
  annote =	{Keywords: chi-bounded, disjointness graph}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.26
On Geometric Set Cover for Orthants

Authors: Karl Bringmann, Sándor Kisfaludi-Bak, Michał Pilipczuk, and Erik Jan van Leeuwen

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We study SET COVER for orthants: Given a set of points in a d-dimensional Euclidean space and a set of orthants of the form (-infty,p_1] x ... x (-infty,p_d], select a minimum number of orthants so that every point is contained in at least one selected orthant. This problem draws its motivation from applications in multi-objective optimization problems. While for d=2 the problem can be solved in polynomial time, for d>2 no algorithm is known that avoids the enumeration of all size-k subsets of the input to test whether there is a set cover of size k. Our contribution is a precise understanding of the complexity of this problem in any dimension d >= 3, when k is considered a parameter: - For d=3, we give an algorithm with runtime n^O(sqrt{k}), thus avoiding exhaustive enumeration. - For d=3, we prove a tight lower bound of n^Omega(sqrt{k}) (assuming ETH). - For d >=slant 4, we prove a tight lower bound of n^Omega(k) (assuming ETH). Here n is the size of the set of points plus the size of the set of orthants. The first statement comes as a corollary of a more general result: an algorithm for SET COVER for half-spaces in dimension 3. In particular, we show that given a set of points U in R^3, a set of half-spaces D in R^3, and an integer k, one can decide whether U can be covered by the union of at most k half-spaces from D in time |D|^O(sqrt{k})* |U|^O(1). We also study approximation for SET COVER for orthants. While in dimension 3 a PTAS can be inferred from existing results, we show that in dimension 4 and larger, there is no 1.05-approximation algorithm with runtime f(k)* n^o(k) for any computable f, where k is the optimum.
Cite as

Karl Bringmann, Sándor Kisfaludi-Bak, Michał Pilipczuk, and Erik Jan van Leeuwen. On Geometric Set Cover for Orthants. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bringmann_et_al:LIPIcs.ESA.2019.26,
  author =	{Bringmann, Karl and Kisfaludi-Bak, S\'{a}ndor and Pilipczuk, Micha{\l} and van Leeuwen, Erik Jan},
  title =	{{On Geometric Set Cover for Orthants}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.26},
  URN =		{urn:nbn:de:0030-drops-111476},
  doi =		{10.4230/LIPIcs.ESA.2019.26},
  annote =	{Keywords: Set Cover, parameterized complexity, algorithms, Exponential Time Hypothesis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.54
On the Chromatic Number of Disjointness Graphs of Curves

Authors: János Pach and István Tomon

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Let omega(G) and chi(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis. We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that omega(G)=k, then chi(G)<= binom{k+1}{2}. If we only require that every curve is x-monotone and intersects the y-axis, then we have chi(G)<= k+1/2 binom{k+2}{3}. Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist K_k-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Omega(k^{4}) colors. This matches the upper bound up to a constant factor.
Cite as

János Pach and István Tomon. On the Chromatic Number of Disjointness Graphs of Curves. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2019.54,
  author =	{Pach, J\'{a}nos and Tomon, Istv\'{a}n},
  title =	{{On the Chromatic Number of Disjointness Graphs of Curves}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.54},
  URN =		{urn:nbn:de:0030-drops-104582},
  doi =		{10.4230/LIPIcs.SoCG.2019.54},
  annote =	{Keywords: string graph, chromatic number, intersection graph}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.59
Disjointness Graphs of Segments

Authors: János Pach, Gábor Tardos, and Géza Tóth

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

Abstract
The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.
Cite as

János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Segments. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pach_et_al:LIPIcs.SoCG.2017.59,
  author =	{Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za},
  title =	{{Disjointness Graphs of Segments}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{59:1--59:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.59},
  URN =		{urn:nbn:de:0030-drops-71960},
  doi =		{10.4230/LIPIcs.SoCG.2017.59},
  annote =	{Keywords: disjointness graph, chromatic number, clique number, chi-bounded}
}
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