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Documents authored by Colin de Verdière, Éric

Document
DOI: 10.4230/LIPIcs.SoCG.2024.28
Computing Shortest Closed Curves on Non-Orientable Surfaces

Authors: Denys Bulavka, Éric Colin de Verdière, and Niloufar Fuladi

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

Abstract
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orientable ones the classification of non-separating simple closed curves up to ambient homeomorphism is subtler, depending on whether the curve is one-sided or two-sided, and whether it is orienting or not (whether it cuts the surface into an orientable one). We prove that computing a shortest orienting (weakly) simple closed curve on a non-orientable combinatorial surface is NP-hard but fixed-parameter tractable in the genus of the surface. In contrast, we can compute a shortest non-separating non-orienting (weakly) simple closed curve with given sidedness in g^{O(1)} ⋅ n log n time, where g is the genus and n the size of the surface. For these algorithms, we develop tools that can be of independent interest, to compute a variation on canonical systems of loops for non-orientable surfaces based on the computation of an orienting curve, and some covering spaces that are essentially quotients of homology covers.
Cite as

Denys Bulavka, Éric Colin de Verdière, and Niloufar Fuladi. Computing Shortest Closed Curves on Non-Orientable Surfaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bulavka_et_al:LIPIcs.SoCG.2024.28,
  author =	{Bulavka, Denys and Colin de Verdi\`{e}re, \'{E}ric and Fuladi, Niloufar},
  title =	{{Computing Shortest Closed Curves on Non-Orientable Surfaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{28:1--28:16},
  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.2024.28},
  URN =		{urn:nbn:de:0030-drops-199731},
  doi =		{10.4230/LIPIcs.SoCG.2024.28},
  annote =	{Keywords: Surface, Graph, Algorithm, Non-orientable surface}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.32
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.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
DOI: 10.4230/LIPIcs.SoCG.2021
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.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
DOI: 10.4230/LIPIcs.SoCG.2021.0
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
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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.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
DOI: 10.4230/LIPIcs.SoCG.2019.27
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.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
DOI: 10.4230/LIPIcs.STACS.2008.1347
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.
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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.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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