18 Search Results for "Welzl, Emo"


Document
Practical Computation of Graph VC-Dimension

Authors: David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)


Abstract
For any set system ℋ = (V,ℛ), ℛ ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in ℛ. The VC-dimension of ℋ is the size of a largest shattered set in V. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph G = (V,E), the VC-dimension of G is defined as the VC-dimension of (V, N), where N contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ V in G. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the W[1]-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.

Cite as

David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Practical Computation of Graph VC-Dimension. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{coudert_et_al:LIPIcs.SEA.2024.8,
  author =	{Coudert, David and Csik\'{o}s, M\'{o}nika and Ducoffe, Guillaume and Viennot, Laurent},
  title =	{{Practical Computation of Graph VC-Dimension}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.8},
  URN =		{urn:nbn:de:0030-drops-203731},
  doi =		{10.4230/LIPIcs.SEA.2024.8},
  annote =	{Keywords: VC-dimension, graph, algorithm}
}
Document
Track A: Algorithms, Complexity and Games
Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents

Authors: Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the α-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our α-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.

Cite as

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, and Simon Weber. Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{borzechowski_et_al:LIPIcs.ICALP.2024.32,
  author =	{Borzechowski, Michaela and Fearnley, John and Gordon, Spencer and Savani, Rahul and Schnider, Patrick and Weber, Simon},
  title =	{{Two Choices Are Enough for P-LCPs, USOs, and Colorful Tangents}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.32},
  URN =		{urn:nbn:de:0030-drops-201751},
  doi =		{10.4230/LIPIcs.ICALP.2024.32},
  annote =	{Keywords: P-LCP, Unique Sink Orientation, \alpha-Ham Sandwich, search complexity, TFNP, UEOPL}
}
Document
A Canonical Tree Decomposition for Chirotopes

Authors: Mathilde Bouvel, Valentin Feray, Xavier Goaoc, and Florent Koechlin

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


Abstract
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.

Cite as

Mathilde Bouvel, Valentin Feray, Xavier Goaoc, and Florent Koechlin. A Canonical Tree Decomposition for Chirotopes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bouvel_et_al:LIPIcs.SoCG.2024.23,
  author =	{Bouvel, Mathilde and Feray, Valentin and Goaoc, Xavier and Koechlin, Florent},
  title =	{{A Canonical Tree Decomposition for Chirotopes}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{23:1--23:18},
  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.2024.23},
  URN =		{urn:nbn:de:0030-drops-199680},
  doi =		{10.4230/LIPIcs.SoCG.2024.23},
  annote =	{Keywords: Order type, modular decomposition, counting triangulations, mutually avoiding point sets, generating functions, rewriting systems}
}
Document
RANDOM
On Connectivity in Random Graph Models with Limited Dependencies

Authors: Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.

Cite as

Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl. On Connectivity in Random Graph Models with Limited Dependencies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{lengler_et_al:LIPIcs.APPROX/RANDOM.2023.30,
  author =	{Lengler, Johannes and Martinsson, Anders and Petrova, Kalina and Schnider, Patrick and Steiner, Raphael and Weber, Simon and Welzl, Emo},
  title =	{{On Connectivity in Random Graph Models with Limited Dependencies}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{30:1--30:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  URN =		{urn:nbn:de:0030-drops-188556},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.30},
  annote =	{Keywords: Random Graphs, Independence, Dependency, Connectivity, Threshold, Probabilistic Method}
}
Document
Improved Algebraic Degeneracy Testing

Authors: Jean Cardinal and Micha Sharir

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


Abstract
In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers a_1,…,a_k from the set such that F(a_1,…,a_k) = 0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve O^*(n^{k-1}) algorithm for this problem (where the O^*(⋅) notation omits subpolynomial factors). We show that the problem can be solved in time O^*(n^{k - 2 + 4/(k+2)}) for even k and in time O^*(n^{k - 2 + (4k-8)/(k²-5)}) for odd k in the real RAM model of computation. We also prove that for k = 4, the problem can be solved in time O^*(n^2.625) in the algebraic decision tree model, and for k = 5 it can be solved in time O^*(n^3.56) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.

Cite as

Jean Cardinal and Micha Sharir. Improved Algebraic Degeneracy Testing. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{cardinal_et_al:LIPIcs.SoCG.2023.22,
  author =	{Cardinal, Jean and Sharir, Micha},
  title =	{{Improved Algebraic Degeneracy Testing}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{22:1--22:16},
  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.2023.22},
  URN =		{urn:nbn:de:0030-drops-178723},
  doi =		{10.4230/LIPIcs.SoCG.2023.22},
  annote =	{Keywords: Degeneracy testing, k-SUM problem, incidence bounds, Hocroft’s problem, polynomial method, algebraic decision trees}
}
Document
Chains, Koch Chains, and Point Sets with Many Triangulations

Authors: Daniel Rutschmann and Manuel Wettstein

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


Abstract
We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations. We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω(9.08ⁿ) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω(8.65ⁿ) for the maximum number of triangulations of planar point sets.

Cite as

Daniel Rutschmann and Manuel Wettstein. Chains, Koch Chains, and Point Sets with Many Triangulations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{rutschmann_et_al:LIPIcs.SoCG.2022.59,
  author =	{Rutschmann, Daniel and Wettstein, Manuel},
  title =	{{Chains, Koch Chains, and Point Sets with Many Triangulations}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{59:1--59:18},
  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.2022.59},
  URN =		{urn:nbn:de:0030-drops-160678},
  doi =		{10.4230/LIPIcs.SoCG.2022.59},
  annote =	{Keywords: Planar Point Set, Chain, Koch Chain, Triangulation, Maximum Number of Triangulations, Lower Bound}
}
Document
Enclosing Depth and Other Depth Measures

Authors: Patrick Schnider

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

Cite as

Patrick Schnider. Enclosing Depth and Other Depth Measures. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{schnider:LIPIcs.ISAAC.2021.10,
  author =	{Schnider, Patrick},
  title =	{{Enclosing Depth and Other Depth Measures}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.10},
  URN =		{urn:nbn:de:0030-drops-154431},
  doi =		{10.4230/LIPIcs.ISAAC.2021.10},
  annote =	{Keywords: Depth measures, Tukey depth, Tverberg theorem, Combinatorial Geometry}
}
Document
Clustering Under Perturbation Stability in Near-Linear Time

Authors: Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, and Emo Welzl

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is α-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most α. Our main contribution is in presenting efficient exact algorithms for α-stable clustering instances whose running times depend near-linearly on the size of the data set when α ≥ 2 + √3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when α ≥ 2 + √3 + ε for any constant ε > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for α > 5 in any fixed dimension; and for α ≥ 2 + √3 in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.

Cite as

Pankaj K. Agarwal, Hsien-Chih Chang, Kamesh Munagala, Erin Taylor, and Emo Welzl. Clustering Under Perturbation Stability in Near-Linear Time. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{agarwal_et_al:LIPIcs.FSTTCS.2020.8,
  author =	{Agarwal, Pankaj K. and Chang, Hsien-Chih and Munagala, Kamesh and Taylor, Erin and Welzl, Emo},
  title =	{{Clustering Under Perturbation Stability in Near-Linear Time}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.8},
  URN =		{urn:nbn:de:0030-drops-132492},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.8},
  annote =	{Keywords: clustering, stability, local search, dynamic programming, coreset, polyhedral metric, trapezoid decomposition, range query}
}
Document
An Optimal Decentralized (Δ + 1)-Coloring Algorithm

Authors: Daniel Bertschinger, Johannes Lengler, Anders Martinsson, Robert Meier, Angelika Steger, Miloš Trujić, and Emo Welzl

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)


Abstract
Consider the following simple coloring algorithm for a graph on n vertices. Each vertex chooses a color from {1, ..., Δ(G) + 1} uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected O(n log Δ) steps, which is optimal and proves a conjecture of Chakrabarty and de Supinski [SOSA'20].

Cite as

Daniel Bertschinger, Johannes Lengler, Anders Martinsson, Robert Meier, Angelika Steger, Miloš Trujić, and Emo Welzl. An Optimal Decentralized (Δ + 1)-Coloring Algorithm. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 17:1-17:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{bertschinger_et_al:LIPIcs.ESA.2020.17,
  author =	{Bertschinger, Daniel and Lengler, Johannes and Martinsson, Anders and Meier, Robert and Steger, Angelika and Truji\'{c}, Milo\v{s} and Welzl, Emo},
  title =	{{An Optimal Decentralized (\Delta + 1)-Coloring Algorithm}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{17:1--17:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.17},
  URN =		{urn:nbn:de:0030-drops-128837},
  doi =		{10.4230/LIPIcs.ESA.2020.17},
  annote =	{Keywords: Decentralized Algorithm, Distributed Computing, Graph Coloring, Randomized Algorithms}
}
Document
Convex Hulls of Random Order Types

Authors: Xavier Goaoc and Emo Welzl

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


Abstract
We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average 4-8/(n^2 - n +2) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d+o(1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods allow to show the following relative of the Erdős-Szekeres theorem: for any fixed k, as n → ∞, a proportion 1 - O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the 2-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral or one of A₄, S₄ or A₅ (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein.

Cite as

Xavier Goaoc and Emo Welzl. Convex Hulls of Random Order Types. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2020.49,
  author =	{Goaoc, Xavier and Welzl, Emo},
  title =	{{Convex Hulls of Random Order Types}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{49:1--49:15},
  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.2020.49},
  URN =		{urn:nbn:de:0030-drops-122074},
  doi =		{10.4230/LIPIcs.SoCG.2020.49},
  annote =	{Keywords: order type, oriented matroid, Sylvester’s Four-Point Problem, random convex hull, projective plane, excluded pattern, Hadwiger’s transversal theorem, hairy ball theorem}
}
Document
Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips)

Authors: Uli Wagner and Emo Welzl

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


Abstract
Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in P ⧵ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) The bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).

Cite as

Uli Wagner and Emo Welzl. Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{wagner_et_al:LIPIcs.SoCG.2020.67,
  author =	{Wagner, Uli and Welzl, Emo},
  title =	{{Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips)}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{67:1--67:16},
  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.2020.67},
  URN =		{urn:nbn:de:0030-drops-122259},
  doi =		{10.4230/LIPIcs.SoCG.2020.67},
  annote =	{Keywords: triangulation, flip graph, graph connectivity, associahedron, subdivision, convex decomposition, flippable edge, flip complex, regular triangulation, bistellar flip graph, secondary polytope, polyhedral subdivision}
}
Document
The Crossing Tverberg Theorem

Authors: Radoslav Fulek, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner

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


Abstract
The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.

Cite as

Radoslav Fulek, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. The Crossing Tverberg Theorem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{fulek_et_al:LIPIcs.SoCG.2019.38,
  author =	{Fulek, Radoslav and G\"{a}rtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli},
  title =	{{The Crossing Tverberg Theorem}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{38:1--38:13},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.38},
  URN =		{urn:nbn:de:0030-drops-104423},
  doi =		{10.4230/LIPIcs.SoCG.2019.38},
  annote =	{Keywords: Discrete geometry, Tverberg theorem, Crossing Tverberg theorem}
}
Document
Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems

Authors: Jean Cardinal, Jerri Nummenpalo, and Emo Welzl

Published in: LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)


Abstract
We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time O^*(eps^{-0.617}) where eps is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected Theta^*(eps^{-1}), and on all previous algorithms whenever eps = Omega(0.708^n). We also consider algorithms for 3-SAT with an eps fraction of satisfying assignments, and prove that it can be solved in O^*(eps^{-2.27}) deterministic time, and in O^*(eps^{-0.936}) randomized time. Finally, to further demonstrate the applicability of this framework, we also explore how similar techniques can be used for vertex cover problems.

Cite as

Jean Cardinal, Jerri Nummenpalo, and Emo Welzl. Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 11:1-11:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{cardinal_et_al:LIPIcs.IPEC.2017.11,
  author =	{Cardinal, Jean and Nummenpalo, Jerri and Welzl, Emo},
  title =	{{Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems}},
  booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
  pages =	{11:1--11:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-051-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{89},
  editor =	{Lokshtanov, Daniel and Nishimura, Naomi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.11},
  URN =		{urn:nbn:de:0030-drops-85459},
  doi =		{10.4230/LIPIcs.IPEC.2017.11},
  annote =	{Keywords: Satisfiability, Sampling, Parameterized complexity}
}
Document
From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices

Authors: Alexander Pilz, Emo Welzl, and Manuel Wettstein

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


Abstract
A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}. We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently. Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n). Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.

Cite as

Alexander Pilz, Emo Welzl, and Manuel Wettstein. From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{pilz_et_al:LIPIcs.SoCG.2017.54,
  author =	{Pilz, Alexander and Welzl, Emo and Wettstein, Manuel},
  title =	{{From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{54:1--54:16},
  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.2017.54},
  URN =		{urn:nbn:de:0030-drops-72101},
  doi =		{10.4230/LIPIcs.SoCG.2017.54},
  annote =	{Keywords: Geometric Graph, Wheel Set, Simplicial Depth, Gale Transform, Polytope}
}
Document
Order on Order Types

Authors: Alexander Pilz and Emo Welzl

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
Given P and P', equally sized planar point sets in general position, we call a bijection from P to P' crossing-preserving if crossings of connecting segments in P are preserved in P' (extra crossings may occur in P'). If such a mapping exists, we say that P' crossing-dominates P, and if such a mapping exists in both directions, P and P' are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.

Cite as

Alexander Pilz and Emo Welzl. Order on Order Types. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 285-299, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{pilz_et_al:LIPIcs.SOCG.2015.285,
  author =	{Pilz, Alexander and Welzl, Emo},
  title =	{{Order on Order Types}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{285--299},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.285},
  URN =		{urn:nbn:de:0030-drops-51194},
  doi =		{10.4230/LIPIcs.SOCG.2015.285},
  annote =	{Keywords: point set, order type, planar graph, crossing-free geometric graph}
}
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