7 Search Results for "Maria, Clément"

Document
DOI: 10.4230/LIPIcs.SoCG.2023.42
On the Width of Complicated JSJ Decompositions

Authors: Kristóf Huszár and Jonathan Spreer

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

Abstract
Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we show that a "sufficiently complicated" JSJ decomposition of a 3-manifold enforces a "complicated structure" for all of its triangulations. More concretely, we show that, under certain conditions, the treewidth (resp. pathwidth) of the graph that captures the incidences between the pieces of the JSJ decomposition of an irreducible, closed, orientable 3-manifold M yields a linear lower bound on its treewidth tw (M) (resp. pathwidth pw(M)), defined as the smallest treewidth (resp. pathwidth) of the dual graph of any triangulation of M. We present several applications of this result. We give the first example of an infinite family of bounded-treewidth 3-manifolds with unbounded pathwidth. We construct Haken 3-manifolds with arbitrarily large treewidth - previously the existence of such 3-manifolds was only known in the non-Haken case. We also show that the problem of providing a constant-factor approximation for the treewidth (resp. pathwidth) of bounded-degree graphs efficiently reduces to computing a constant-factor approximation for the treewidth (resp. pathwidth) of 3-manifolds.
Cite as

Kristóf Huszár and Jonathan Spreer. On the Width of Complicated JSJ Decompositions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{huszar_et_al:LIPIcs.SoCG.2023.42,
  author =	{Husz\'{a}r, Krist\'{o}f and Spreer, Jonathan},
  title =	{{On the Width of Complicated JSJ Decompositions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{42:1--42:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.42},
  URN =		{urn:nbn:de:0030-drops-178920},
  doi =		{10.4230/LIPIcs.SoCG.2023.42},
  annote =	{Keywords: computational 3-manifold topology, fixed-parameter tractability, generalized Heegaard splittings, JSJ decompositions, pathwidth, treewidth, triangulations}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.78
Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds

Authors: Clément Maria and Owen Rouillé

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements. This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations. In this paper, we propose a new method to find a triangulation that admits a solution to the gluing equations, using convex optimization and localized combinatorial modifications. It is based on Casson and Rivin’s reformulation of the equations. We provide a novel approach to modify a triangulation and update its geometry, along with experimental results to support the new method.
Cite as

Clément Maria and Owen Rouillé. Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{maria_et_al:LIPIcs.ESA.2022.78,
  author =	{Maria, Cl\'{e}ment and Rouill\'{e}, Owen},
  title =	{{Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{78:1--78:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.78},
  URN =		{urn:nbn:de:0030-drops-170168},
  doi =		{10.4230/LIPIcs.ESA.2022.78},
  annote =	{Keywords: knots and 3-manifolds, triangulation, hyperbolic structure, Thurston equations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.53
Parameterized Complexity of Quantum Knot Invariants

Authors: Clément Maria

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

Abstract
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by planar diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a O(N^{3/2 cw} poly(n)) ∈ N^O(√n) time algorithm to compute any Reshetikhin-Turaev invariant - derived from a simple Lie algebra 𝔤 - of a link presented by a planar diagram with n crossings and carving-width cw, and whose components are coloured with 𝔤-modules of dimension at most N. For example, this includes the N^{th}-coloured Jones polynomial.
Cite as

Clément Maria. Parameterized Complexity of Quantum Knot Invariants. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{maria:LIPIcs.SoCG.2021.53,
  author =	{Maria, Cl\'{e}ment},
  title =	{{Parameterized Complexity of Quantum Knot Invariants}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{53:1--53:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.53},
  URN =		{urn:nbn:de:0030-drops-138527},
  doi =		{10.4230/LIPIcs.SoCG.2021.53},
  annote =	{Keywords: computational knot theory, parameterized complexity, quantum invariants}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.56
Intrinsic Topological Transforms via the Distance Kernel Embedding

Authors: Clément Maria, Steve Oudot, and Elchanan Solomon

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

Abstract
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms for abstract metric measure spaces. Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator. To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms. We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds.
Cite as

Clément Maria, Steve Oudot, and Elchanan Solomon. Intrinsic Topological Transforms via the Distance Kernel Embedding. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{maria_et_al:LIPIcs.SoCG.2020.56,
  author =	{Maria, Cl\'{e}ment and Oudot, Steve and Solomon, Elchanan},
  title =	{{Intrinsic Topological Transforms via the Distance Kernel Embedding}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{56:1--56:15},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.56},
  URN =		{urn:nbn:de:0030-drops-122145},
  doi =		{10.4230/LIPIcs.SoCG.2020.56},
  annote =	{Keywords: Topological Transforms, Persistent Homology, Inverse Problems, Spectral Geometry, Algebraic Topology, Topological Data Analysis}
}
Document
DOI: 10.4230/OASIcs.Tokenomics.2019.6
A Smart Contract Oracle for Approximating Real-World, Real Number Values

Authors: William George and Clément Lesaege

Published in: OASIcs, Volume 71, International Conference on Blockchain Economics, Security and Protocols (Tokenomics 2019)

Abstract
A key challenge of smart contract systems is the fact that many useful contracts require access to information that does not natively live on the blockchain. While miners can verify the value of a hash or the validity of a digital signature, they cannot determine who won an election, whether there is a flood in Paris, or even what is the price of ether in US dollars, even though this information might be necessary to execute prediction market, insurance, or financial contracts respectively. A number of promising projects and research developments have provided a better understanding of how one might construct a decentralized, binary oracle - namely an oracle that can respond by one of two possibilities, typically "yes" or "no", even while not requiring the interaction of a trusted third party. In this work, we extend these ideas to construct a general-purpose, decentralized oracle that can estimate the value of a real-world quantity that is in a dense totally ordered set, such as R. In particular, this proposal can be used to estimate real number valued quantities, such as required for a price oracle. We will establish a number of desirable properties about this proposal. Particularly, we will see that the precision of the output is tunable to users' needs.
Cite as

William George and Clément Lesaege. A Smart Contract Oracle for Approximating Real-World, Real Number Values. In International Conference on Blockchain Economics, Security and Protocols (Tokenomics 2019). Open Access Series in Informatics (OASIcs), Volume 71, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{george_et_al:OASIcs.Tokenomics.2019.6,
  author =	{George, William and Lesaege, Cl\'{e}ment},
  title =	{{A Smart Contract Oracle for Approximating Real-World, Real Number Values}},
  booktitle =	{International Conference on Blockchain Economics, Security and Protocols (Tokenomics 2019)},
  pages =	{6:1--6:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-108-5},
  ISSN =	{2190-6807},
  year =	{2020},
  volume =	{71},
  editor =	{Danos, Vincent and Herlihy, Maurice and Potop-Butucaru, Maria and Prat, Julien and Tucci-Piergiovanni, Sara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.Tokenomics.2019.6},
  URN =		{urn:nbn:de:0030-drops-119705},
  doi =		{10.4230/OASIcs.Tokenomics.2019.6},
  annote =	{Keywords: price oracle, Ethereum, blockchain}
}
Document
DOI: 10.4230/LIPIcs.AofA.2018.18
Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

Authors: Michael Drmota, Lander Ramos, Clément Requilé, and Juanjo Rué

Published in: LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

Abstract
We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in series-parallel graphs (and related graph classes) with n vertices. In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988]. We also show that these two parameters converge to a central limit law.
Cite as

Michael Drmota, Lander Ramos, Clément Requilé, and Juanjo Rué. Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{drmota_et_al:LIPIcs.AofA.2018.18,
  author =	{Drmota, Michael and Ramos, Lander and Requil\'{e}, Cl\'{e}ment and Ru\'{e}, Juanjo},
  title =	{{Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes}},
  booktitle =	{29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{Fill, James Allen and Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.18},
  URN =		{urn:nbn:de:0030-drops-89117},
  doi =		{10.4230/LIPIcs.AofA.2018.18},
  annote =	{Keywords: Asymptotic enumeration, central limit laws, subcritical graph classes, maximal independent set, maximal matching}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.64
Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants

Authors: Clément Maria and Jonathan Spreer

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold. In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds. The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants.
Cite as

Clément Maria and Jonathan Spreer. Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{maria_et_al:LIPIcs.ESA.2016.64,
  author =	{Maria, Cl\'{e}ment and Spreer, Jonathan},
  title =	{{Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.64},
  URN =		{urn:nbn:de:0030-drops-64050},
  doi =		{10.4230/LIPIcs.ESA.2016.64},
  annote =	{Keywords: low-dimensional topology, triangulations of 3-manifolds, cohomology theory, Turaev-Viro invariants, combinatorial algorithms}
}
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