8 Search Results for "Colin de Verdière, Éric"


Volume

LIPIcs, Volume 189

37th International Symposium on Computational Geometry (SoCG 2021)

SoCG 2021, June 7-11, 2021, Buffalo, NY, USA (Virtual Conference)

Editors: Kevin Buchin and Éric Colin de Verdière

Document
An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes

Authors: Éric Colin de Verdière and Thomas Magnard

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)


Abstract
We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.

Cite as

Éric Colin de Verdière and Thomas Magnard. An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{colindeverdiere_et_al:LIPIcs.ESA.2021.32,
  author =	{Colin de Verdi\`{e}re, \'{E}ric and Magnard, Thomas},
  title =	{{An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{32:1--32:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus 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.2021.32},
  URN =		{urn:nbn:de:0030-drops-146139},
  doi =		{10.4230/LIPIcs.ESA.2021.32},
  annote =	{Keywords: computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability}
}
Document
Complete Volume
LIPIcs, Volume 189, SoCG 2021, Complete Volume

Authors: Kevin Buchin and Éric Colin de Verdière

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


Abstract
LIPIcs, Volume 189, SoCG 2021, Complete Volume

Cite as

37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 1-978, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@Proceedings{buchin_et_al:LIPIcs.SoCG.2021,
  title =	{{LIPIcs, Volume 189, SoCG 2021, Complete Volume}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{1--978},
  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},
  URN =		{urn:nbn:de:0030-drops-137987},
  doi =		{10.4230/LIPIcs.SoCG.2021},
  annote =	{Keywords: LIPIcs, Volume 189, SoCG 2021, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Kevin Buchin and Éric Colin de Verdière

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


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

Cite as

37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{buchin_et_al:LIPIcs.SoCG.2021.0,
  author =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{0:i--0:xviii},
  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.2021.0},
  URN =		{urn:nbn:de:0030-drops-137993},
  doi =		{10.4230/LIPIcs.SoCG.2021.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs

Authors: Vincent Cohen-Addad, Éric Colin de Verdière, Dániel Marx, and Arnaud de Mesmay

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


Abstract
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We prove a time lower bound for this problem of n^{Omega(g/log g)} conditionally to ETH. In other words, the first n^{O(g)}-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n^{Omega(sqrt{gt + g^2}/log(gt))}, conditionally to ETH, for any choice of the genus g >=0 of the graph and the number of terminals t >=4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.

Cite as

Vincent Cohen-Addad, Éric Colin de Verdière, Dániel Marx, and Arnaud de Mesmay. Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2019.27,
  author =	{Cohen-Addad, Vincent and Colin de Verdi\`{e}re, \'{E}ric and Marx, D\'{a}niel and de Mesmay, Arnaud},
  title =	{{Almost Tight Lower Bounds for Hard Cutting Problems in Embedded Graphs}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{27:1--27:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.27},
  URN =		{urn:nbn:de:0030-drops-104311},
  doi =		{10.4230/LIPIcs.SoCG.2019.27},
  annote =	{Keywords: Cut graph, Multiway cut, Surface, Lower bound, Parameterized Complexity, Exponential Time Hypothesis}
}
Document
Shortest k-Disjoint Paths via Determinants

Authors: Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)


Abstract
The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-disjoint paths between k specified pairs of vertices (s_i, t_i) in a given graph, if they exist. The decision version of the shortest k-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly, the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest k-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdière and Schrijver [Éric Colin de Verdière and Alexander Schrijver, 2011] for the general one-face problem. We do so by providing a randomised NC^2 algorithm along with an O(n^{omega/2}) time randomised sequential algorithm, for any fixed k. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. The algorithms are based on a bijection between a shortest k-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "pure" cycle covers.

Cite as

Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee. Shortest k-Disjoint Paths via Determinants. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2018.19,
  author =	{Datta, Samir and Iyer, Siddharth and Kulkarni, Raghav and Mukherjee, Anish},
  title =	{{Shortest k-Disjoint Paths via Determinants}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{19:1--19:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.19},
  URN =		{urn:nbn:de:0030-drops-99183},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.19},
  annote =	{Keywords: disjoint paths, planar graph, parallel algorithm, cycle cover, determinant, inclusion-exclusion}
}
Document
Embedding Graphs into Two-Dimensional Simplicial Complexes

Authors: Éric Colin de Verdière, Thomas Magnard, and Bojan Mohar

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


Abstract
We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general. The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed. Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).

Cite as

Éric Colin de Verdière, Thomas Magnard, and Bojan Mohar. Embedding Graphs into Two-Dimensional Simplicial Complexes. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{verdiere_et_al:LIPIcs.SoCG.2018.27,
  author =	{Verdi\`{e}re, \'{E}ric Colin de and Magnard, Thomas and Mohar, Bojan},
  title =	{{Embedding Graphs into Two-Dimensional Simplicial Complexes}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{27:1--27:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.27},
  URN =		{urn:nbn:de:0030-drops-87401},
  doi =		{10.4230/LIPIcs.SoCG.2018.27},
  annote =	{Keywords: computational topology, embedding, simplicial complex, graph, surface}
}
Document
Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs

Authors: Éric Colin de Verdiére and Alexander Schrijver

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)


Abstract
Let $G$ be a directed planar graph of complexity~$n$, each arc having a nonnegative length. Let $s$ and~$t$ be two distinct faces of~$G$; let $s_1,ldots,s_k$ be vertices incident with~$s$; let $t_1,ldots,t_k$ be vertices incident with~$t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in~$G$, with minimal total length, in $O(knlog n)$ time.

Cite as

Éric Colin de Verdiére and Alexander Schrijver. Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 181-192, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


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@InProceedings{colindeverdiere_et_al:LIPIcs.STACS.2008.1347,
  author =	{Colin de Verdi\'{e}re, \'{E}ric and Schrijver, Alexander},
  title =	{{Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{181--192},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1347},
  URN =		{urn:nbn:de:0030-drops-13474},
  doi =		{10.4230/LIPIcs.STACS.2008.1347},
  annote =	{Keywords: Algorithm, planar graph, disjoint paths, shortest path}
}
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