13 Search Results for "Keller, Chaya"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.61
Complements of Finite Unions of Convex Sets

Authors: Chaya Keller and Micha A. Perles

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions - i.e., sets of the form S = ℝ^d ⧵ (∪_{i=1}^n K_i), where K_i are convex sets. In the first part of the paper we study isolated points in S, whose number is related to the Betti numbers of ∪_{i=1}^n K_i and to its non-convexity properties. We obtain upper bounds on the number of such points, which are sharp for n = 3 and significantly improve previous bounds of Lawrence and Morris (2009) for all n ≪ 2^d/d. In the second part of the paper we study coverings of S by well-behaved sets. We show that S can be covered by at most g(d,n) flats of different dimensions, in such a way that each x ∈ S is covered by a flat whose dimension equals the "local dimension" of S in the neighborhood of x. Furthermore, we determine the structure of a minimum cover that satisfies this property. Then, we study quantitative aspects of this minimum cover and obtain sharp upper bounds on its size in various settings.
Cite as

Chaya Keller and Micha A. Perles. Complements of Finite Unions of Convex Sets. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{keller_et_al:LIPIcs.SoCG.2026.61,
  author =	{Keller, Chaya and Perles, Micha A.},
  title =	{{Complements of Finite Unions of Convex Sets}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.61},
  URN =		{urn:nbn:de:0030-drops-258684},
  doi =		{10.4230/LIPIcs.SoCG.2026.61},
  annote =	{Keywords: convexity, unions of convex sets}
}
Document
Research
DOI: 10.4230/TGDK.3.3.4
Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web

Authors: Florian Ruosch, Cristina Sarasua, and Abraham Bernstein

Published in: TGDK, Volume 3, Issue 3 (2025). Transactions on Graph Data and Knowledge, Volume 3, Issue 3

Abstract
In Argument Mining, predicting argumentative relations between texts (or spans) remains one of the most challenging aspects, even more so in the cross-document setting. This paper makes three key contributions to advance research in this domain. We first extend an existing dataset, the Sci-Arg corpus, by annotating it with explicit inter-document argumentative relations, thereby allowing arguments to be distributed over several documents forming an Argument Web; these new annotations are published using Semantic Web technologies (RDF, OWL). Second, we explore and evaluate three automated approaches for predicting these inter-document argumentative relations, establishing critical baselines on the new dataset. We find that a simple classifier based on discourse indicators with access to context outperforms neural methods. Third, we conduct a comparative analysis of these approaches for both intra- and inter-document settings, identifying statistically significant differences in results that indicate the necessity of distinguishing between these two scenarios. Our findings highlight significant challenges in this complex domain and open crucial avenues for future research on the Argument Web of Science, particularly for those interested in leveraging Semantic Web technologies and knowledge graphs to understand scholarly discourse. With this, we provide the first stepping stones in the form of a benchmark dataset, three baseline methods, and an initial analysis for a systematic exploration of this field relevant to the Web of Data and Science.
Cite as

Florian Ruosch, Cristina Sarasua, and Abraham Bernstein. Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web. In Transactions on Graph Data and Knowledge (TGDK), Volume 3, Issue 3, pp. 4:1-4:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{ruosch_et_al:TGDK.3.3.4,
  author =	{Ruosch, Florian and Sarasua, Cristina and Bernstein, Abraham},
  title =	{{Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{4:1--4:33},
  ISSN =	{2942-7517},
  year =	{2025},
  volume =	{3},
  number =	{3},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.3.3.4},
  URN =		{urn:nbn:de:0030-drops-252159},
  doi =		{10.4230/TGDK.3.3.4},
  annote =	{Keywords: Argument Mining, Large Language Models, Knowledge Graphs, Link Prediction}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.21
On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

Authors: Parinya Chalermsook, Ly Orgo, and Minoo Zarsav

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
This paper considers the Zarankiewicz problem in bipartite graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Let Z(n;k) be the maximum number of edges in a bipartite graph with n nodes and is free of a k-by-k biclique. Note that Z(n;k) ∈ Ω(nk) for all "natural" graph classes. Our first result reveals a separation between bipartite graphs of Ferrers dimension three and four: while we show that Z(n;k) ≤ 9n(k-1) for graphs of Ferrers dimension three, Z(n;k) ∈ Ω(n k ⋅ (log n)/(log log n)) for Ferrers dimension four graphs (Chan & Har-Peled, 2023) (Chazelle, 1990). To complement this, we derive a tight upper bound of 2n(k-1) for chordal bipartite graphs and 54n(k-1) for grid intersection graphs (GIG), a prominent graph class residing in four Ferrers dimensions and capturing planar bipartite graphs as well as bipartite intersection graphs of rectangles. Previously, the best-known bound for GIG was Z(n;k) ∈ O(2^{O(k)} n), implied by the results of Fox & Pach (2006) and Mustafa & Pach (2016). Our results advance and offer new insights into the interplay between Ferrers dimensions and extremal combinatorics.
Cite as

Parinya Chalermsook, Ly Orgo, and Minoo Zarsav. On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chalermsook_et_al:LIPIcs.GD.2025.21,
  author =	{Chalermsook, Parinya and Orgo, Ly and Zarsav, Minoo},
  title =	{{On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.21},
  URN =		{urn:nbn:de:0030-drops-250074},
  doi =		{10.4230/LIPIcs.GD.2025.21},
  annote =	{Keywords: Bipartite graph classes, extremal graph theory, geometric intersection graphs, Zarankiewicz problem, bicliques}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.33
On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects

Authors: Timothy M. Chan, Chaya Keller, and Shakhar Smorodinsky

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
In this paper we study the hypergraph Zarankiewicz’s problem in a geometric setting - for r-partite intersection hypergraphs of families of geometric objects. Our main results are essentially sharp bounds for families of axis-parallel boxes in ℝ^d and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound O_{d,t}(n^{r-1}((log n)/(log log n))^{d-1}). The best previous bound was larger by a factor of about (log n)^{d(2^{r-1}-2)}. For pseudo-discs, we obtain the bound O_t(n^{r-1}(log n)^{r-2}), which is sharp up to logarithmic factors. As this hypergraph has no algebraic structure, no improvement of Erdős' 60-year-old O(n^{r-(1/t^{r-1})}) bound was known for this setting. Futhermore, even in the special case of discs for which the semialgebraic structure can be used, our result improves the best known result by a factor of Ω̃(n^{(2r-2)/(3r-2)}). To obtain our results, we use the recently improved results for the graph Zarankiewicz’s problem in the corresponding settings, along with a variety of combinatorial and geometric techniques, including shallow cuttings, biclique covers, transversals, and planarity.
Cite as

Timothy M. Chan, Chaya Keller, and Shakhar Smorodinsky. On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2025.33,
  author =	{Chan, Timothy M. and Keller, Chaya and Smorodinsky, Shakhar},
  title =	{{On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{33:1--33:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.33},
  URN =		{urn:nbn:de:0030-drops-231850},
  doi =		{10.4230/LIPIcs.SoCG.2025.33},
  annote =	{Keywords: Zarankiewicz’s Problem, hypergraphs, intersection graphs, axis-parallel boxes, pseudo-discs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.61
k-Dimensional Transversals for Fat Convex Sets

Authors: Attila Jung and Dömötör Pálvölgyi

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We prove a fractional Helly theorem for k-flats intersecting fat convex sets. A family ℱ of sets is said to be ρ-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by ρ. We prove that for every dimension d and positive reals ρ and α there exists a positive β = β(d,ρ, α) such that if ℱ is a finite family of ρ-fat convex sets in ℝ^d and an α-fraction of the (k+2)-size subfamilies from ℱ can be hit by a k-flat, then there is a k-flat that intersects at least a β-fraction of the sets of ℱ. We prove spherical and colorful variants of the above results and prove a (p,k+2)-theorem for k-flats intersecting balls.
Cite as

Attila Jung and Dömötör Pálvölgyi. k-Dimensional Transversals for Fat Convex Sets. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 61:1-61:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jung_et_al:LIPIcs.SoCG.2025.61,
  author =	{Jung, Attila and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
  title =	{{k-Dimensional Transversals for Fat Convex Sets}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{61:1--61:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.61},
  URN =		{urn:nbn:de:0030-drops-232136},
  doi =		{10.4230/LIPIcs.SoCG.2025.61},
  annote =	{Keywords: discrete geometry, transversals, Helly, hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.19
Polychromatic Coloring of Tuples in Hypergraphs

Authors: Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
A hypergraph H consists of a set V of vertices and a set E of hyperedges that are subsets of V. A t-tuple of H is a subset of t vertices of V. A t-tuple k-coloring of H is a mapping of its t-tuples into k colors. A coloring is called (t,k,f)-polychromatic if each hyperedge of E that has at least f vertices contains tuples of all the k colors. Let f_H(t,k) be the minimum f such that H has a (t,k,f)-polychromatic coloring. For a family of hypergraphs ℋ let f_H(t,k) be the maximum f_H(t,k) over all hypergraphs H in H. Determining f_H(t,k) has been an active research direction in recent years. This is challenging even for t = 1. We present several new results in this direction for t ≥ 2. - Let H be the family of hypergraphs H that is obtained by taking any set P of points in ℝ², setting V: = P and E: = {d ∩ P: d is a disk in ℝ²}. We prove that f_ H(2,k) ≤ 3.7^k, that is, the pairs of points (2-tuples) can be k-colored such that any disk containing at least 3.7^k points has pairs of all colors. We generalize this result to points and balls in higher dimensions. - For the family H of hypergraphs that are defined by grid vertices and axis-parallel rectangles in the plane, we show that f_H(2,k) ≤ √{ck ln k} for some constant c. We then generalize this to higher dimensions, to other shapes, and to tuples of larger size. - For the family H of shrinkable hypergraphs of VC-dimension at most d we prove that f_ H(d+1,k) ≤ c^k for some constant c = c(d). Towards this bound, we obtain a result of independent interest: Every hypergraph with n vertices and with VC-dimension at most d has a (d+1)-tuple T of depth at least n/c, i.e., any hyperedge that contains T also contains n/c other vertices. - For the relationship between t-tuple coloring and vertex coloring in any hypergraph H we establish the inequality 1/e⋅ tk^{1/t} ≤ f_H(t,k) ≤ f_H(1,tk^{1/t}). For the special case of k = 2, referred to as the bichromatic coloring, we prove that t+1 ≤ f_H(t,2) ≤ max{f_H(1,2), t+1}; this improves upon the previous best known upper bound. - We study the relationship between tuple coloring and epsilon nets. In particular we show that if f_H(1,k) = O(k) for a hypergraph H with n vertices, then for any 0 < ε < 1 the t-tuples of H can be partitioned into Ω((εn/t)^t) ε-t-nets. This bound is tight when t is a constant.
Cite as

Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković. Polychromatic Coloring of Tuples in Hypergraphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2025.19,
  author =	{Biniaz, Ahmad and De Carufel, Jean-Lou and Maheshwari, Anil and Smid, Michiel and Smorodinsky, Shakhar and Stojakovi\'{c}, Milo\v{s}},
  title =	{{Polychromatic Coloring of Tuples in Hypergraphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.19},
  URN =		{urn:nbn:de:0030-drops-231718},
  doi =		{10.4230/LIPIcs.SoCG.2025.19},
  annote =	{Keywords: Hypergraph Coloring, Polychromatic Coloring, Geometric Hypergraphs, Cover Decomposable Hypergraphs, Epsilon Nets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.66
Zarankiewicz’s Problem via ε-t-Nets

Authors: Chaya Keller and Shakhar Smorodinsky

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
The classical Zarankiewicz’s problem asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph K_{t,t}. Kővári, Sós and Turán proved an upper bound of O(n^{2-1/t}). Fox et al. obtained an improved bound of O(n^{2-1/d}) for graphs of VC-dimension d (where d < t). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. Chan and Har-Peled further improved Basit et al.’s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of O(n log log n) for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz’s problem, via ε-t-nets - a recently introduced generalization of the classical notion of ε-nets. Using the new approach, we obtain a sharp bound of O(n) for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the O(n^{2-1/d}) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n {log n}/{log log n}) bound for the intersection graph of two families of axis-parallel rectangles.
Cite as

Chaya Keller and Shakhar Smorodinsky. Zarankiewicz’s Problem via ε-t-Nets. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{keller_et_al:LIPIcs.SoCG.2024.66,
  author =	{Keller, Chaya and Smorodinsky, Shakhar},
  title =	{{Zarankiewicz’s Problem via \epsilon-t-Nets}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{66:1--66:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.66},
  URN =		{urn:nbn:de:0030-drops-200111},
  doi =		{10.4230/LIPIcs.SoCG.2024.66},
  annote =	{Keywords: Zarankiewicz’s Problem, \epsilon-t-nets, pseudo-discs, VC-dimension}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.33
A Solution to Ringel’s Circle Problem

Authors: James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak

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

Abstract
We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.
Cite as

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{davies_et_al:LIPIcs.SoCG.2022.33,
  author =	{Davies, James and Keller, Chaya and Kleist, Linda and Smorodinsky, Shakhar and Walczak, Bartosz},
  title =	{{A Solution to Ringel’s Circle Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{33:1--33:14},
  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.2022.33},
  URN =		{urn:nbn:de:0030-drops-160413},
  doi =		{10.4230/LIPIcs.SoCG.2022.33},
  annote =	{Keywords: circle arrangement, chromatic number, Gallai’s theorem, polynomial method}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.50
An (ℵ₀,k+2)-Theorem for k-Transversals

Authors: Chaya Keller and Micha A. Perles

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

Abstract
A family ℱ of sets satisfies the (p,q)-property if among every p members of ℱ, some q can be pierced by a single point. The celebrated (p,q)-theorem of Alon and Kleitman asserts that for any p ≥ q ≥ d+1, any family ℱ of compact convex sets in ℝ^d that satisfies the (p,q)-property can be pierced by a finite number c(p,q,d) of points. A similar theorem with respect to piercing by (d-1)-dimensional flats, called (d-1)-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an (ℵ₀,k+2)-theorem with respect to k-transversals: Let ℱ be an infinite family of sets in ℝ^d such that each A ∈ ℱ contains a ball of radius r and is contained in a ball of radius R, and let 0 ≤ k < d. If among every ℵ₀ elements of ℱ, some k+2 can be pierced by a k-dimensional flat, then ℱ can be pierced by a finite number of k-dimensional flats. This is the first (p,q)-theorem in which the assumption is weakened to an (∞,⋅) assumption. Our proofs combine geometric and topological tools.
Cite as

Chaya Keller and Micha A. Perles. An (ℵ₀,k+2)-Theorem for k-Transversals. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{keller_et_al:LIPIcs.SoCG.2022.50,
  author =	{Keller, Chaya and Perles, Micha A.},
  title =	{{An (\aleph₀,k+2)-Theorem for k-Transversals}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{50:1--50:14},
  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.2022.50},
  URN =		{urn:nbn:de:0030-drops-160581},
  doi =		{10.4230/LIPIcs.SoCG.2022.50},
  annote =	{Keywords: convexity, (p,q)-theorem, k-transversal, infinite (p,q)-theorem}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICALP.2021.4
Error Resilient Space Partitioning (Invited Talk)

Authors: Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, and Ran J. Tessler

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
In this paper we consider a new type of space partitioning which bridges the gap between continuous and discrete spaces in an error resilient way. It is motivated by the problem of rounding noisy measurements from some continuous space such as ℝ^d to a discrete subset of representative values, in which each tile in the partition is defined as the preimage of one of the output points. Standard rounding schemes seem to be inherently discontinuous across tile boundaries, but in this paper we show how to make it perfectly consistent (with error resilience ε) by guaranteeing that any pair of consecutive measurements X₁ and X₂ whose L₂ distance is bounded by ε will be rounded to the same nearby representative point in the discrete output space. We achieve this resilience by allowing a few bits of information about the first measurement X₁ to be unidirectionally communicated to and used by the rounding process of the second measurement X₂. Minimizing this revealed information can be particularly important in privacy-sensitive applications such as COVID-19 contact tracing, in which we want to find out all the cases in which two persons were at roughly the same place at roughly the same time, by comparing cryptographically hashed versions of their itineraries in an error resilient way. The main problem we study in this paper is characterizing the achievable tradeoffs between the amount of information provided and the error resilience for various dimensions. We analyze the problem by considering the possible colored tilings of the space with k available colors, and use the color of the tile in which X₁ resides as the side information. We obtain our upper and lower bounds with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Sperner’s lemma, and Čech cohomology. In particular, we show that when X_i ∈ ℝ^d, communicating log₂(d+1) bits of information is both sufficient and necessary (in the worst case) to achieve positive resilience, and when d=3 we obtain a tight upper and lower asymptotic bound of (0.561 …)k^{1/3} on the achievable error resilience when we provide log₂(k) bits of information about X₁’s color.
Cite as

Orr Dunkelman, Zeev Geyzel, Chaya Keller, Nathan Keller, Eyal Ronen, Adi Shamir, and Ran J. Tessler. Error Resilient Space Partitioning (Invited Talk). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dunkelman_et_al:LIPIcs.ICALP.2021.4,
  author =	{Dunkelman, Orr and Geyzel, Zeev and Keller, Chaya and Keller, Nathan and Ronen, Eyal and Shamir, Adi and Tessler, Ran J.},
  title =	{{Error Resilient Space Partitioning}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{4:1--4:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.4},
  URN =		{urn:nbn:de:0030-drops-140731},
  doi =		{10.4230/LIPIcs.ICALP.2021.4},
  annote =	{Keywords: space partition, high-dimensional rounding, error resilience, sphere packing, Sperner’s lemma, Brunn-Minkowski theorem, \v{C}ech cohomology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.47
No Krasnoselskii Number for General Sets

Authors: Chaya Keller and Micha A. Perles

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

Abstract
For a family ℱ of non-empty sets in ℝ^d, the Krasnoselskii number of ℱ is the smallest m such that for any S ∈ ℱ, if every m or fewer points of S are visible from a common point in S, then any finite subset of S is visible from a single point. More than 35 years ago, Peterson asked whether there exists a Krasnoselskii number for general sets in ℝ^d. The best known positive result is Krasnoselskii number 3 for closed sets in the plane, and the best known negative result is that if a Krasnoselskii number for general sets in ℝ^d exists, it cannot be smaller than (d+1)². In this paper we answer Peterson’s question in the negative by showing that there is no Krasnoselskii number for the family of all sets in ℝ². The proof is non-constructive, and uses transfinite induction and the well-ordering theorem. In addition, we consider Krasnoselskii numbers with respect to visibility through polygonal paths of length ≤ n, for which an analogue of Krasnoselskii’s theorem for compact simply connected sets was proved by Magazanik and Perles. We show, by an explicit construction, that for any n ≥ 2, there is no Krasnoselskii number for the family of compact sets in ℝ² with respect to visibility through paths of length ≤ n. (Here the counterexamples are finite unions of line segments.)
Cite as

Chaya Keller and Micha A. Perles. No Krasnoselskii Number for General Sets. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 47:1-47:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{keller_et_al:LIPIcs.SoCG.2021.47,
  author =	{Keller, Chaya and Perles, Micha A.},
  title =	{{No Krasnoselskii Number for General Sets}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{47:1--47:11},
  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.2021.47},
  URN =		{urn:nbn:de:0030-drops-138462},
  doi =		{10.4230/LIPIcs.SoCG.2021.47},
  annote =	{Keywords: visibility, Helly-type theorems, Krasnoselskii’s theorem, transfinite induction, well-ordering theorem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.5
The ε-t-Net Problem

Authors: Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky

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

Abstract
We study a natural generalization of the classical ε-net problem (Haussler - Welzl 1987), which we call the ε-t-net problem: Given a hypergraph on n vertices and parameters t and ε ≥ t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ε n contains a set in S. When t=1, this corresponds to the ε-net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an ε-t-net of size O((1+log t)d/ε log 1/ε). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/ε)-sized ε-t-nets. We also present an explicit construction of ε-t-nets (including ε-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ε-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.
Cite as

Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky. The ε-t-Net Problem. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{alon_et_al:LIPIcs.SoCG.2020.5,
  author =	{Alon, Noga and Jartoux, Bruno and Keller, Chaya and Smorodinsky, Shakhar and Yuditsky, Yelena},
  title =	{{The \epsilon-t-Net Problem}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{5:1--5:15},
  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.2020.5},
  URN =		{urn:nbn:de:0030-drops-121639},
  doi =		{10.4230/LIPIcs.SoCG.2020.5},
  annote =	{Keywords: epsilon-nets, geometric hypergraphs, VC-dimension, linear union complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.51
From a (p,2)-Theorem to a Tight (p,q)-Theorem

Authors: Chaya Keller and Shakhar Smorodinsky

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

Abstract
A family F of sets is said to satisfy the (p,q)-property if among any p sets of F some q have a non-empty intersection. The celebrated (p,q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in R^d that satisfies the (p,q)-property for some q >= d+1, can be pierced by a fixed number (independent on the size of the family) f_d(p,q) of points. The minimum such piercing number is denoted by {HD}_d(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever q > (d-1)/d p+1 the piercing number is {HD}_d(p,q)=p-q+1; no exact values of {HD}_d(p,q) were found ever since. While for an arbitrary family of compact convex sets in R^d, d >= 2, a (p,2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol'nikov (1972) used a (p,2)-theorem for axis-parallel rectangles to show that {HD}_{rect}(p,q)=p-q+1 holds for all q>sqrt{2p}. These are the only values of q for which {HD}_{rect}(p,q) is known exactly. In this paper we present a general method which allows using a (p,2)-theorem as a bootstrapping to obtain a tight (p,q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we show that {HD}_{d-box}(p,q)=p-q+1 holds for all q > c' log^{d-1} p, and in particular, {HD}_{rect}(p,q)=p-q+1 holds for all q >= 7 log_2 p (compared to q >= sqrt{2p}, obtained by Wegner and Dol'nikov more than 40 years ago). In addition, for several classes of families, we present improved (p,2)-theorems, some of which can be used as a bootstrapping to obtain tight (p,q)-theorems. In particular, we show that any family F of compact convex sets in R^d with Helly number 2 admits a (p,2)-theorem with piercing number O(p^{2d-1}), and thus, satisfies {HD}_{F}(p,q)=p-q+1 for all q>cp^{1-1/(2d-1)}, for a universal constant c.
Cite as

Chaya Keller and Shakhar Smorodinsky. From a (p,2)-Theorem to a Tight (p,q)-Theorem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{keller_et_al:LIPIcs.SoCG.2018.51,
  author =	{Keller, Chaya and Smorodinsky, Shakhar},
  title =	{{From a (p,2)-Theorem to a Tight (p,q)-Theorem}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{51:1--51:14},
  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.2018.51},
  URN =		{urn:nbn:de:0030-drops-87640},
  doi =		{10.4230/LIPIcs.SoCG.2018.51},
  annote =	{Keywords: (p,q)-Theorem, convexity, transversals, (p,2)-theorem, axis-parallel rectangles}
}
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