81 Search Results for "Chambers, Erin"

Volume

LIPIcs, Volume 258

39th International Symposium on Computational Geometry (SoCG 2023)

SoCG 2023, June 12-15, 2023, Dallas, Texas, USA

Editors: Erin W. Chambers and Joachim Gudmundsson

Document
DOI: 10.4230/LIPIcs.SoCG.2024.69
The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms

Authors: Hana Dal Poz Kouřimská, André Lieutier, and Mathijs Wintraecken

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

Abstract
We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.
Cite as

Hana Dal Poz Kouřimská, André Lieutier, and Mathijs Wintraecken. The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 69:1-69:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dalpozkourimska_et_al:LIPIcs.SoCG.2024.69,
  author =	{Dal Poz Kou\v{r}imsk\'{a}, Hana and Lieutier, Andr\'{e} and Wintraecken, Mathijs},
  title =	{{The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{69:1--69:18},
  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.2024.69},
  URN =		{urn:nbn:de:0030-drops-200149},
  doi =		{10.4230/LIPIcs.SoCG.2024.69},
  annote =	{Keywords: Medial axis, Hausdorff distance, Lipschitz continuity}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.SoCG.2023
LIPIcs, Volume 258, SoCG 2023, Complete Volume

Authors: Erin W. Chambers and Joachim Gudmundsson

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
LIPIcs, Volume 258, SoCG 2023, Complete Volume
Cite as

39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 1-1058, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{chambers_et_al:LIPIcs.SoCG.2023,
  title =	{{LIPIcs, Volume 258, SoCG 2023, Complete Volume}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{1--1058},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023},
  URN =		{urn:nbn:de:0030-drops-178498},
  doi =		{10.4230/LIPIcs.SoCG.2023},
  annote =	{Keywords: LIPIcs, Volume 258, SoCG 2023, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.SoCG.2023.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Erin W. Chambers and Joachim Gudmundsson

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chambers_et_al:LIPIcs.SoCG.2023.0,
  author =	{Chambers, Erin W. and Gudmundsson, Joachim},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{0:i--0:xx},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.0},
  URN =		{urn:nbn:de:0030-drops-178501},
  doi =		{10.4230/LIPIcs.SoCG.2023.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.1
Geometric Embeddability of Complexes Is ∃ℝ-Complete

Authors: Mikkel Abrahamsen, Linda Kleist, and Tillmann Miltzow

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ℝ^d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability.
Cite as

Mikkel Abrahamsen, Linda Kleist, and Tillmann Miltzow. Geometric Embeddability of Complexes Is ∃ℝ-Complete. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.1,
  author =	{Abrahamsen, Mikkel and Kleist, Linda and Miltzow, Tillmann},
  title =	{{Geometric Embeddability of Complexes Is \exists\mathbb{R}-Complete}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{1:1--1:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.1},
  URN =		{urn:nbn:de:0030-drops-178518},
  doi =		{10.4230/LIPIcs.SoCG.2023.1},
  annote =	{Keywords: simplicial complex, geometric embedding, linear embedding, hypergraph, recognition, existential theory of the reals}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.2
Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks

Authors: Mikkel Abrahamsen and Bartosz Walczak

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar.
Cite as

Mikkel Abrahamsen and Bartosz Walczak. Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2023.2,
  author =	{Abrahamsen, Mikkel and Walczak, Bartosz},
  title =	{{Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{2:1--2:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.2},
  URN =		{urn:nbn:de:0030-drops-178523},
  doi =		{10.4230/LIPIcs.SoCG.2023.2},
  annote =	{Keywords: geometric intersection graph, convex disk, homothet, similarity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.3
Lower Bounds for Intersection Reporting Among Flat Objects

Authors: Peyman Afshani and Pingan Cheng

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Recently, Ezra and Sharir [Esther Ezra and Micha Sharir, 2022] showed an O(n^{3/2+σ}) space and O(n^{1/2+σ}) query time data structure for ray shooting among triangles in ℝ³. This improves the upper bound given by the classical S(n)Q(n)⁴ = O(n^{4+σ}) space-time tradeoff for the first time in almost 25 years and in fact lies on the tradeoff curve of S(n)Q(n)³ = O(n^{3+σ}). However, it seems difficult to apply their techniques beyond this specific space and time combination. This pheonomenon appears persistently in almost all recent advances of flat object intersection searching, e.g., line-tetrahedron intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], triangle-triangle intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], or even among flat semialgebraic objects [Agarwal et al., 2022]. We give a timely explanation to this phenomenon from a lower bound perspective. We prove that given a set 𝒮 of (d-1)-dimensional simplicies in ℝ^d, any data structure that can report all intersections with a query line in small (n^o(1)) query time must use Ω(n^{2(d-1)-o(1)}) space. This dashes the hope of any significant improvement to the tradeoff curves for small query time and almost matches the classical upper bound. We also obtain an almost matching space lower bound of Ω(n^{6-o(1)}) for triangle-triangle intersection reporting in ℝ⁴ when the query time is small. Along the way, we further develop the previous lower bound techniques by Afshani and Cheng [Afshani and Cheng, 2021; Afshani and Cheng, 2022].
Cite as

Peyman Afshani and Pingan Cheng. Lower Bounds for Intersection Reporting Among Flat Objects. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2023.3,
  author =	{Afshani, Peyman and Cheng, Pingan},
  title =	{{Lower Bounds for Intersection Reporting Among Flat Objects}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.3},
  URN =		{urn:nbn:de:0030-drops-178536},
  doi =		{10.4230/LIPIcs.SoCG.2023.3},
  annote =	{Keywords: Computational Geometry, Intersection Searching, Data Structure Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.4
Computing Instance-Optimal Kernels in Two Dimensions

Authors: Pankaj K. Agarwal and Sariel Har-Peled

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Let P be a set of n points in ℝ². For a parameter ε ∈ (0,1), a subset C ⊆ P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weak ε-kernel of P if its directional width approximates that of P in every direction. Let 𝗄_ε(P) (resp. 𝗄^𝗐_ε(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(n 𝗄_ε(P)log n)-time algorithm for computing an ε-kernel of P of size 𝗄_ε(P), and an O(n²log n)-time algorithm for computing a weak ε-kernel of P of size 𝗄^𝗐_ε(P). We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of ε-core, a convex polygon lying inside ch(P), prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.
Cite as

Pankaj K. Agarwal and Sariel Har-Peled. Computing Instance-Optimal Kernels in Two Dimensions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.4,
  author =	{Agarwal, Pankaj K. and Har-Peled, Sariel},
  title =	{{Computing Instance-Optimal Kernels in Two Dimensions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.4},
  URN =		{urn:nbn:de:0030-drops-178544},
  doi =		{10.4230/LIPIcs.SoCG.2023.4},
  annote =	{Keywords: Coreset, approximation, kernel}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.5
Line Intersection Searching Amid Unit Balls in 3-Space

Authors: Pankaj K. Agarwal and Esther Ezra

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Let ℬ be a set of n unit balls in ℝ³. We present a linear-size data structure for storing ℬ that can determine in O^*(n^{1/2}) time whether a query line intersects any ball of ℬ and report all k such balls in additional O(k) time. The data structure can be constructed in O(n log n) time. (The O^*(⋅) notation hides subpolynomial factors, e.g., of the form O(n^ε), for arbitrarily small ε > 0, and their coefficients which depend on ε.) We also consider the dual problem: Let ℒ be a set of n lines in ℝ³. We preprocess ℒ, in O^*(n²) time, into a data structure of size O^*(n²) that can determine in O^*(1) time whether a query unit ball intersects any line of ℒ, or report all k such lines in additional O(k) time.
Cite as

Pankaj K. Agarwal and Esther Ezra. Line Intersection Searching Amid Unit Balls in 3-Space. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.5,
  author =	{Agarwal, Pankaj K. and Ezra, Esther},
  title =	{{Line Intersection Searching Amid Unit Balls in 3-Space}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.5},
  URN =		{urn:nbn:de:0030-drops-178559},
  doi =		{10.4230/LIPIcs.SoCG.2023.5},
  annote =	{Keywords: Intersection searching, cylindrical range searching, partition trees, union of cylinders}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.6
Drawings of Complete Multipartite Graphs up to Triangle Flips

Authors: Oswin Aichholzer, Man-Kwun Chiu, Hung P. Hoang, Michael Hoffmann, Jan Kynčl, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan’s Theorem states that for any two simple drawings of the complete graph K_n with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n^{16}). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph K_{m,n} minus two edges and K_{m,n} plus one edge for any m,n ≥ 4, as well as K_n minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges.
Cite as

Oswin Aichholzer, Man-Kwun Chiu, Hung P. Hoang, Michael Hoffmann, Jan Kynčl, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger. Drawings of Complete Multipartite Graphs up to Triangle Flips. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2023.6,
  author =	{Aichholzer, Oswin and Chiu, Man-Kwun and Hoang, Hung P. and Hoffmann, Michael and Kyn\v{c}l, Jan and Maus, Yannic and Vogtenhuber, Birgit and Weinberger, Alexandra},
  title =	{{Drawings of Complete Multipartite Graphs up to Triangle Flips}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.6},
  URN =		{urn:nbn:de:0030-drops-178563},
  doi =		{10.4230/LIPIcs.SoCG.2023.6},
  annote =	{Keywords: Simple drawings, simple topological graphs, complete graphs, multipartite graphs, k-partite graphs, bipartite graphs, Gioan’s Theorem, triangle flips, Reidemeister moves}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.7
Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

Authors: Ángel Javier Alonso and Michael Kerber

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.
Cite as

Ángel Javier Alonso and Michael Kerber. Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2023.7,
  author =	{Alonso, \'{A}ngel Javier and Kerber, Michael},
  title =	{{Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.7},
  URN =		{urn:nbn:de:0030-drops-178570},
  doi =		{10.4230/LIPIcs.SoCG.2023.7},
  annote =	{Keywords: Multiparameter persistent homology, Clustering, Decomposition of persistence modules, Elder Rule}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.8
On Helly Numbers of Exponential Lattices

Authors: Gergely Ambrus, Martin Balko, Nóra Frankl, Attila Jung, and Márton Naszódi

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Given a set S ⊆ ℝ², define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family ℱ of convex sets in ℝ² such that the intersection of any N or fewer members of ℱ contains at least one point of S, there is a point of S common to all members of ℱ. We prove that the Helly numbers of exponential lattices {αⁿ : n ∈ ℕ₀}² are finite for every α > 1 and we determine their exact values in some instances. In particular, we obtain H({2ⁿ : n ∈ ℕ₀}²) = 5, solving a problem posed by Dillon (2021). For real numbers α, β > 1, we also fully characterize exponential lattices L(α,β) = {αⁿ : n ∈ ℕ₀} × {βⁿ : n ∈ ℕ₀} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_α(β) is rational.
Cite as

Gergely Ambrus, Martin Balko, Nóra Frankl, Attila Jung, and Márton Naszódi. On Helly Numbers of Exponential Lattices. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ambrus_et_al:LIPIcs.SoCG.2023.8,
  author =	{Ambrus, Gergely and Balko, Martin and Frankl, N\'{o}ra and Jung, Attila and Nasz\'{o}di, M\'{a}rton},
  title =	{{On Helly Numbers of Exponential Lattices}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.8},
  URN =		{urn:nbn:de:0030-drops-178584},
  doi =		{10.4230/LIPIcs.SoCG.2023.8},
  annote =	{Keywords: Helly numbers, exponential lattices, Diophantine approximation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.9
Optimal Volume-Sensitive Bounds for Polytope Approximation

Authors: Sunil Arya and David M. Mount

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body K of diameter Δ in ℝ^d for fixed d. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that Θ((Δ/ε)^{(d-1)/2}) vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies. A natural way to characterize a convex object’s skinniness is in terms of its relationship to the Euclidean ball. Given a convex body K, define its volume diameter Δ_d to be the diameter of a Euclidean ball of the same volume as K, and define its surface diameter Δ_{d-1} analogously for surface area. It follows from generalizations of the isoperimetric inequality that Δ ≥ Δ_{d-1} ≥ Δ_d. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to O((Δ_{d-1}/ε)^{(d-1)/2}). In this paper, we strengthen this by proving the existence of an approximation with O((Δ_d/ε)^{(d-1)/2}) facets. This improvement is a result of the combination of a number of new ideas. As in prior work, we exploit properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the following more general problem. Given two convex bodies, one nested within the other, find a low-complexity convex polytope that is sandwiched between them. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.
Cite as

Sunil Arya and David M. Mount. Optimal Volume-Sensitive Bounds for Polytope Approximation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arya_et_al:LIPIcs.SoCG.2023.9,
  author =	{Arya, Sunil and Mount, David M.},
  title =	{{Optimal Volume-Sensitive Bounds for Polytope Approximation}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.9},
  URN =		{urn:nbn:de:0030-drops-178592},
  doi =		{10.4230/LIPIcs.SoCG.2023.9},
  annote =	{Keywords: Approximation algorithms, convexity, Macbeath regions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.10
Coresets for Clustering in Geometric Intersection Graphs

Authors: Sayan Bandyapadhyay, Fedor V. Fomin, and Tanmay Inamdar

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
Designing coresets - small-space sketches of the data preserving cost of the solutions within (1± ε)-approximate factor - is an important research direction in the study of center-based k-clustering problems, such as k-means or k-median. Feldman and Langberg [STOC'11] have shown that for k-clustering of n points in general metrics, it is possible to obtain coresets whose size depends logarithmically in n. Moreover, such a dependency in n is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independent of n for special metrics, like d-dimensional Euclidean space [Huang, Vishnoi, STOC'20], doubling metrics [Huang, Jiang, Li, Wu, FOCS'18], metrics of graphs of bounded treewidth [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML’20], or graphs excluding a fixed minor [Braverman, Jiang, Krauthgamer, Wu, SODA’21]. In this paper, we provide the first constructions of coresets whose size does not depend on n for k-clustering in the metrics induced by geometric intersection graphs. For example, we obtain (k log²k)/ε^𝒪(1) size coresets for k-clustering in Euclidean-weighted unit-disk graphs (UDGs) and unit-square graphs (USGs). These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining coresets whose size is independent of n. The proof of our theorem builds on the recent work of Cohen-Addad, Saulpic, and Schwiegelshohn [STOC '21], which ensures small-sized coresets conditioned on the existence of an interesting set of centers, called centroid set. The main technical contribution of our work is the proof of the existence of such a small-sized centroid set for graphs that satisfy the two canonical properties. Loosely speaking, the metrics of geometric intersection graphs are "similar" to the Euclidean metrics for points that are close, and to the shortest path metrics of planar graphs for points that are far apart. The main technical challenge in constructing centroid sets of small sizes is in combining these two very different metrics. The new coreset construction helps to design the first (1+ε)-approximation for center-based clustering problems in UDGs and USGs, that is fixed-parameter tractable in k and ε (FPT-AS).
Cite as

Sayan Bandyapadhyay, Fedor V. Fomin, and Tanmay Inamdar. Coresets for Clustering in Geometric Intersection Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.10,
  author =	{Bandyapadhyay, Sayan and Fomin, Fedor V. and Inamdar, Tanmay},
  title =	{{Coresets for Clustering in Geometric Intersection Graphs}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.10},
  URN =		{urn:nbn:de:0030-drops-178605},
  doi =		{10.4230/LIPIcs.SoCG.2023.10},
  annote =	{Keywords: k-median, k-means, clustering, coresets, geometric graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.11
Minimum-Membership Geometric Set Cover, Revisited

Authors: Sayan Bandyapadhyay, William Lochet, Saket Saurabh, and Jie Xue

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We revisit a natural variant of the geometric set cover problem, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set S of points and a set ℛ of geometric objects, and the goal is to find a subset ℛ^* ⊆ ℛ to cover all points in S such that the membership of S with respect to ℛ^*, denoted by memb(S,ℛ^*), is minimized, where memb(S,ℛ^*) = max_{p ∈ S} |{R ∈ ℛ^*: p ∈ R}|. We give the first polynomial-time approximation algorithms for MMGSC in ℝ². Specifically, we achieve the following two main results. - We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time n^{O(opt)} where opt is the optimum of the problem (i.e., the minimum membership). - We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of o(log n) in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find ℛ^* ⊆ ℛ to cover S such that the ply of ℛ^* is minimized, where the ply is defined as the maximum number of objects in ℛ^* which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in O(n^{12}) time. We give a significantly simpler constant-approximation algorithm with near-linear running time.
Cite as

Sayan Bandyapadhyay, William Lochet, Saket Saurabh, and Jie Xue. Minimum-Membership Geometric Set Cover, Revisited. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.11,
  author =	{Bandyapadhyay, Sayan and Lochet, William and Saurabh, Saket and Xue, Jie},
  title =	{{Minimum-Membership Geometric Set Cover, Revisited}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.11},
  URN =		{urn:nbn:de:0030-drops-178610},
  doi =		{10.4230/LIPIcs.SoCG.2023.11},
  annote =	{Keywords: geometric set cover, geometric optimization, approximation algorithms}
}
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