42 Search Results for "Bonnet, Édouard"


Document
Treewidth Is NP-Complete on Cubic Graphs

Authors: Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
In this paper, we show that Treewidth is NP-complete for cubic graphs, thereby improving the result by Bodlaender and Thilikos from 1997 that Treewidth is NP-complete on graphs with maximum degree at most 9. We add a new and simpler proof of the NP-completeness of treewidth, and show that Treewidth remains NP-complete on subcubic induced subgraphs of the infinite 3-dimensional grid.

Cite as

Hans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, and Ondřej Suchý. Treewidth Is NP-Complete on Cubic Graphs. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 7:1-7:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2023.7,
  author =	{Bodlaender, Hans L. and Bonnet, \'{E}douard and Jaffke, Lars and Knop, Du\v{s}an and Lima, Paloma T. and Milani\v{c}, Martin and Ordyniak, Sebastian and Pandey, Sukanya and Such\'{y}, Ond\v{r}ej},
  title =	{{Treewidth Is NP-Complete on Cubic Graphs}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{7:1--7:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.7},
  URN =		{urn:nbn:de:0030-drops-194263},
  doi =		{10.4230/LIPIcs.IPEC.2023.7},
  annote =	{Keywords: Treewidth, cubic graphs, degree, NP-completeness}
}
Document
Stretch-Width

Authors: Édouard Bonnet and Julien Duron

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This leaves open the possibility of efficient algorithms for a broad fragment of problems within Monadic Second-Order (MSO) logic on graphs of bounded stretch-width. In this direction, we prove that graphs of bounded maximum degree and bounded stretch-width have at most logarithmic treewidth. As a consequence, in classes of bounded stretch-width, Maximum Independent Set can be solved in subexponential time 2^{Õ(n^{8/9})} on n-vertex graphs, and, if further the maximum degree is bounded, Existential Counting Modal Logic [Pilipczuk '11] can be model-checked in polynomial time. We also give a polynomial-time O(OPT²)-approximation for the stretch-width of symmetric 0,1-matrices or ordered graphs. Somewhat unexpectedly, we prove that exponential subdivisions of bounded-degree graphs have bounded stretch-width. This allows to complement the logarithmic upper bound of treewidth with a matching lower bound. We leave as open the existence of an efficient approximation algorithm for the stretch-width of unordered graphs, if the exponential subdivisions of all graphs have bounded stretch-width, and if graphs of bounded stretch-width have logarithmic clique-width (or rank-width).

Cite as

Édouard Bonnet and Julien Duron. Stretch-Width. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 8:1-8:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2023.8,
  author =	{Bonnet, \'{E}douard and Duron, Julien},
  title =	{{Stretch-Width}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{8:1--8:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.8},
  URN =		{urn:nbn:de:0030-drops-194279},
  doi =		{10.4230/LIPIcs.IPEC.2023.8},
  annote =	{Keywords: Contraction sequences, twin-width, clique-width, algorithms, algorithmic metatheorems}
}
Document
Twin-Width of Graphs with Tree-Structured Decompositions

Authors: Irene Heinrich and Simon Raßmann

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 [Édouard Bonnet et al., 2022; Édouard Bonnet et al., 2020], a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of [Édouard Bonnet and Hugues Déprés, 2023; Édouard Bonnet and Hugues Déprés, 2022], which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain optimal linear bounds on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.

Cite as

Irene Heinrich and Simon Raßmann. Twin-Width of Graphs with Tree-Structured Decompositions. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 25:1-25:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{heinrich_et_al:LIPIcs.IPEC.2023.25,
  author =	{Heinrich, Irene and Ra{\ss}mann, Simon},
  title =	{{Twin-Width of Graphs with Tree-Structured Decompositions}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.25},
  URN =		{urn:nbn:de:0030-drops-194449},
  doi =		{10.4230/LIPIcs.IPEC.2023.25},
  annote =	{Keywords: twin-width, quasi-4 connected components, strong tree-width}
}
Document
PACE Solver Description
PACE Solver Description: Exact (GUTHMI) and Heuristic (GUTHM)

Authors: Alexander Leonhardt, Holger Dell, Anselm Haak, Frank Kammer, Johannes Meintrup, Ulrich Meyer, and Manuel Penschuck

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
Twin-width (tww) is a parameter measuring the similarity of an undirected graph to a co-graph [Édouard Bonnet et al., 2022]. It is useful to analyze the parameterized complexity of various graph problems. This paper presents two algorithms to compute the twin-width and to provide a contraction sequence as witness. The two algorithms are motivated by the PACE 2023 challenge, one for the exact track and one for the heuristic track. Each algorithm produces a contraction sequence witnessing (i) the minimal twin-width admissible by the graph in the exact track (ii) an upper bound on the twin-width as tight as possible in the heuristic track. Our heuristic algorithm relies on several greedy approaches with different performance characteristics to find and improve solutions. For large graphs we use locality sensitive hashing to approximately identify suitable contraction candidates. The exact solver follows a branch-and-bound design. It relies on the heuristic algorithm to provide initial upper bounds, and uses lower bounds via contraction sequences to show the optimality of a heuristic solution found in some branch.

Cite as

Alexander Leonhardt, Holger Dell, Anselm Haak, Frank Kammer, Johannes Meintrup, Ulrich Meyer, and Manuel Penschuck. PACE Solver Description: Exact (GUTHMI) and Heuristic (GUTHM). In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 37:1-37:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{leonhardt_et_al:LIPIcs.IPEC.2023.37,
  author =	{Leonhardt, Alexander and Dell, Holger and Haak, Anselm and Kammer, Frank and Meintrup, Johannes and Meyer, Ulrich and Penschuck, Manuel},
  title =	{{PACE Solver Description: Exact (GUTHMI) and Heuristic (GUTHM)}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{37:1--37:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.37},
  URN =		{urn:nbn:de:0030-drops-194563},
  doi =		{10.4230/LIPIcs.IPEC.2023.37},
  annote =	{Keywords: PACE 2023 Challenge, Heuristic, Exact, Twin-Width}
}
Document
PACE Solver Description
PACE Solver Description: RedAlert - Heuristic Track

Authors: Édouard Bonnet and Julien Duron

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)


Abstract
We present RedAlert, a heuristic solver for twin-width, submitted to the Heuristic Track of the 2023 edition of the Parameterized Algorithms and Computational Experiments (PACE) challenge.

Cite as

Édouard Bonnet and Julien Duron. PACE Solver Description: RedAlert - Heuristic Track. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 40:1-40:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2023.40,
  author =	{Bonnet, \'{E}douard and Duron, Julien},
  title =	{{PACE Solver Description: RedAlert - Heuristic Track}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{40:1--40:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.40},
  URN =		{urn:nbn:de:0030-drops-194591},
  doi =		{10.4230/LIPIcs.IPEC.2023.40},
  annote =	{Keywords: twin-width, contraction sequences, heuristic, pair sampling, pair filtering}
}
Document
Maximum Independent Set When Excluding an Induced Minor: K₁ + tK₂ and tC₃ ⊎ C₄

Authors: Édouard Bonnet, Julien Duron, Colin Geniet, Stéphan Thomassé, and Alexandra Wesolek

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
Dallard, Milanič, and Štorgel [arXiv '22] ask if, for every class excluding a fixed planar graph H as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when H is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when H is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the t-vertex cycle, C_t [Gartland et al., STOC '21], and the disjoint union of t triangles, tC₃ [Bonamy et al., SODA '23]. We give, for every integer t, a polynomial-time algorithm running in n^O(t⁵) when H is the friendship graph K₁ + tK₂ (t disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in n^{O(t² log n) + f(t)}, with f a single-exponential function, when H is tC₃ ⊎ C₄ (the disjoint union of t triangles and a 4-vertex cycle). The former generalizes the algorithm readily obtained from Alekseev’s structural result on graphs excluding tK₂ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.’s result.

Cite as

Édouard Bonnet, Julien Duron, Colin Geniet, Stéphan Thomassé, and Alexandra Wesolek. Maximum Independent Set When Excluding an Induced Minor: K₁ + tK₂ and tC₃ ⊎ C₄. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bonnet_et_al:LIPIcs.ESA.2023.23,
  author =	{Bonnet, \'{E}douard and Duron, Julien and Geniet, Colin and Thomass\'{e}, St\'{e}phan and Wesolek, Alexandra},
  title =	{{Maximum Independent Set When Excluding an Induced Minor: K₁ + tK₂ and tC₃ ⊎ C₄}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.23},
  URN =		{urn:nbn:de:0030-drops-186769},
  doi =		{10.4230/LIPIcs.ESA.2023.23},
  annote =	{Keywords: Maximum Independent Set, forbidden induced minors, quasipolynomial-time algorithms}
}
Document
First Order Logic and Twin-Width in Tournaments

Authors: Colin Geniet and Stéphan Thomassé

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs. The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments.

Cite as

Colin Geniet and Stéphan Thomassé. First Order Logic and Twin-Width in Tournaments. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 53:1-53:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{geniet_et_al:LIPIcs.ESA.2023.53,
  author =	{Geniet, Colin and Thomass\'{e}, St\'{e}phan},
  title =	{{First Order Logic and Twin-Width in Tournaments}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{53:1--53:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.53},
  URN =		{urn:nbn:de:0030-drops-187061},
  doi =		{10.4230/LIPIcs.ESA.2023.53},
  annote =	{Keywords: Tournaments, twin-width, first-order logic, model checking, NIP, small classes}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Authors: Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth. Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class C of graphs is flip-flat if for every fixed radius r, every sufficiently large set of vertices of a graph G ∈ C contains a large subset of vertices with mutual distance larger than r, where the distance is measured in some graph G' that can be obtained from G by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.

Cite as

Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk. Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 125:1-125:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{dreier_et_al:LIPIcs.ICALP.2023.125,
  author =	{Dreier, Jan and M\"{a}hlmann, Nikolas and Siebertz, Sebastian and Toru\'{n}czyk, Szymon},
  title =	{{Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{125:1--125:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.125},
  URN =		{urn:nbn:de:0030-drops-181779},
  doi =		{10.4230/LIPIcs.ICALP.2023.125},
  annote =	{Keywords: stability, NIP, combinatorial characterization, first-order model checking}
}
Document
Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width

Authors: Pierre Bergé, Édouard Bonnet, Hugues Déprés, and Rémi Watrigant

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)


Abstract
For any ε > 0, we give a polynomial-time n^ε-approximation algorithm for Max Independent Set in graphs of bounded twin-width given with an O(1)-sequence. This result is derived from the following time-approximation trade-off: We establish an O(1)^{2^q-1}-approximation algorithm running in time exp(O_q(n^{2^{-q}})), for every integer q ⩾ 0. Guided by the same framework, we obtain similar approximation algorithms for Min Coloring and Max Induced Matching. In general graphs, all these problems are known to be highly inapproximable: for any ε > 0, a polynomial-time n^{1-ε}-approximation for any of them would imply that P=NP [Håstad, FOCS '96; Zuckerman, ToC '07; Chalermsook et al., SODA '13]. We generalize the algorithms for Max Independent Set and Max Induced Matching to the independent (induced) packing of any fixed connected graph H. In contrast, we show that such approximation guarantees on graphs of bounded twin-width given with an O(1)-sequence are very unlikely for Min Independent Dominating Set, and somewhat unlikely for Longest Path and Longest Induced Path. Regarding the existence of better approximation algorithms, there is a (very) light evidence that the obtained approximation factor of n^ε for Max Independent Set may be best possible. This is the first in-depth study of the approximability of problems in graphs of bounded twin-width. Prior to this paper, essentially the only such result was a polynomial-time O(1)-approximation algorithm for Min Dominating Set [Bonnet et al., ICALP '21].

Cite as

Pierre Bergé, Édouard Bonnet, Hugues Déprés, and Rémi Watrigant. Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 10:1-10:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{berge_et_al:LIPIcs.STACS.2023.10,
  author =	{Berg\'{e}, Pierre and Bonnet, \'{E}douard and D\'{e}pr\'{e}s, Hugues and Watrigant, R\'{e}mi},
  title =	{{Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.10},
  URN =		{urn:nbn:de:0030-drops-176629},
  doi =		{10.4230/LIPIcs.STACS.2023.10},
  annote =	{Keywords: Approximation algorithms, bounded twin-width}
}
Document
Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication

Authors: Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, and Stéphan Thomassé

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)


Abstract
We continue developing the theory around the twin-width of totally ordered binary structures (or equivalently, matrices over a finite alphabet), initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class (i.e., a set of matrices closed under taking submatrices) has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable (FPT) algorithm for first-order model checking on structures given with an O(1)-sequence (certificate of bounded twin-width) and the fact that first-order transductions of bounded twin-width classes have bounded twin-width, both established in Twin-width I, extend to first-order logic with modular counting quantifiers. We make explicit a win-win argument obtained as a by-product of Twin-width IV, and somewhat similar to bidimensionality, that we call rank-bidimensionality. This generalizes the seminal work of Guillemot and Marx [SODA '14], which builds on the Marcus-Tardos theorem [JCTA '04]. It works on general matrices (not only on classes of bounded twin-width) and, for example, yields FPT algorithms deciding if a small matrix is a parity or a linear minor of another matrix given in input, or exactly computing the grid or mixed number of a given matrix (i.e., the maximum integer k such that the row set and the column set of the matrix can be partitioned into k intervals, with each of the k² defined cells containing a non-zero entry, or two distinct rows and two distinct columns, respectively). Armed with the above-mentioned extension to modular counting, we show that the twin-width of the product of two conformal matrices A, B (i.e., whose dimensions are such that AB is defined) over a finite field is bounded by a function of the twin-width of A, of B, and of the size of the field. Furthermore, if A and B are n × n matrices of twin-width d over F_q, we show that AB can be computed in time O_{d,q}(n² log n). We finally present an ad hoc algorithm to efficiently multiply two matrices of bounded twin-width, with a single-exponential dependence in the twin-width bound. More precisely, pipelined to observations and results of Pilipczuk et al. [STACS '22], we obtain the following. If the inputs are given in a compact tree-like form (witnessing twin-width at most d), called twin-decomposition of width d, then two n × n matrices A, B over F₂ can be multiplied in time 4^{d+o(d)}n, in the sense that a twin-decomposition of their product AB, with width 2^{d+o(d)}, is output within that time, and each entry of AB can be queried in time O_d(log log n). Furthermore, for every ε > 0, the query time can be brought to constant time O(1/ε) if the running time is increased to near-linear 4^{d+o(d)}n^{1+ε}. Notably, the running time is sublinear (essentially square root) in the number of (non-zero) entries.

Cite as

Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, and Stéphan Thomassé. Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bonnet_et_al:LIPIcs.STACS.2023.15,
  author =	{Bonnet, \'{E}douard and Giocanti, Ugo and Ossona de Mendez, Patrice and Thomass\'{e}, St\'{e}phan},
  title =	{{Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.15},
  URN =		{urn:nbn:de:0030-drops-176675},
  doi =		{10.4230/LIPIcs.STACS.2023.15},
  annote =	{Keywords: Twin-width, matrices, parity and linear minors, model theory, linear algebra, matrix multiplication, algorithms, computational complexity}
}
Document
Twin-Width VIII: Delineation and Win-Wins

Authors: Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, and Stéphan Thomassé

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)


Abstract
We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures of subclasses D of C, FO model checking on D is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that K_{t,t}-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H₄-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated. More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with O(1)-sequences [BKTW, JACM '22], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If p is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then p(G) ⩾ k can be decided in FPT time f(k) ⋅ |V(G)|^O(1). For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width.

Cite as

Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, and Stéphan Thomassé. Twin-Width VIII: Delineation and Win-Wins. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 9:1-9:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2022.9,
  author =	{Bonnet, \'{E}douard and Chakraborty, Dibyayan and Kim, Eun Jung and K\"{o}hler, Noleen and Lopes, Raul and Thomass\'{e}, St\'{e}phan},
  title =	{{Twin-Width VIII: Delineation and Win-Wins}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.9},
  URN =		{urn:nbn:de:0030-drops-173650},
  doi =		{10.4230/LIPIcs.IPEC.2022.9},
  annote =	{Keywords: Twin-width, intersection graphs, visibility graphs, monadic dependence and stability, first-order model checking}
}
Document
Weighted Model Counting with Twin-Width

Authors: Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)


Abstract
Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula F along with a bound k and asks for the weighted sum of all models with at most k positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of F plus k. We show that this result is tight: it is neither possible to drop the bound k nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

Cite as

Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider. Weighted Model Counting with Twin-Width. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 15:1-15:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ganian_et_al:LIPIcs.SAT.2022.15,
  author =	{Ganian, Robert and Pokr\'{y}vka, Filip and Schidler, Andr\'{e} and Simonov, Kirill and Szeider, Stefan},
  title =	{{Weighted Model Counting with Twin-Width}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.15},
  URN =		{urn:nbn:de:0030-drops-166896},
  doi =		{10.4230/LIPIcs.SAT.2022.15},
  annote =	{Keywords: Weighted model counting, twin-width, parameterized complexity, SAT}
}
Document
Track A: Algorithms, Complexity and Games
Deciding Twin-Width at Most 4 Is NP-Complete

Authors: Pierre Bergé, Édouard Bonnet, and Hugues Déprés

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)


Abstract
We show that determining if an n-vertex graph has twin-width at most 4 is NP-complete, and requires time 2^Ω(n/log n) unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that n-vertex graphs subdivided at least 2 log n times have twin-width at most 4. We also show how to encode trigraphs H (2-edge colored graphs involved in the definition of twin-width) into graphs G, in the sense that every d-sequence (sequence of vertex contractions witnessing that the twin-width is at most d) of G inevitably creates H as an induced subtrigraph, whereas there exists a partial d-sequence that actually goes from G to H. We believe that these facts and their proofs can be of independent interest.

Cite as

Pierre Bergé, Édouard Bonnet, and Hugues Déprés. Deciding Twin-Width at Most 4 Is NP-Complete. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 18:1-18:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{berge_et_al:LIPIcs.ICALP.2022.18,
  author =	{Berg\'{e}, Pierre and Bonnet, \'{E}douard and D\'{e}pr\'{e}s, Hugues},
  title =	{{Deciding Twin-Width at Most 4 Is NP-Complete}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.18},
  URN =		{urn:nbn:de:0030-drops-163595},
  doi =		{10.4230/LIPIcs.ICALP.2022.18},
  annote =	{Keywords: Twin-width, lower bounds}
}
Document
Twin-Width and Polynomial Kernels

Authors: Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant

Published in: LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)


Abstract
We study the existence of polynomial kernels for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. It was previously observed in [Bonnet et al., ICALP'21] that the problem k-Independent Set allows no polynomial kernel on graph of bounded twin-width by a very simple argument, which extends to several other problems such as k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching. In this work, we examine the k-Dominating Set and variants of k-Vertex Cover for the existence of polynomial kernels. As a main result, we show that k-Dominating Set does not admit a polynomial kernel on graphs of twin-width at most 4 under a standard complexity-theoretic assumption. The reduction is intricate, especially due to the effort to bring the twin-width down to 4, and it can be tweaked to work for Connected k-Dominating Set and Total k-Dominating Set with a slightly worse bound on the twin-width. On the positive side, we obtain a simple quadratic vertex kernel for Connected k-Vertex Cover and Capacitated k-Vertex Cover on graphs of bounded twin-width. These kernels rely on that graphs of bounded twin-width have Vapnik-Chervonenkis (VC) density 1, that is, for any vertex set X, the number of distinct neighborhoods in X is at most c⋅|X|, where c is a constant depending only on the twin-width. Interestingly the kernel applies to any graph class of VC density 1, and does not require a witness sequence. We also present a more intricate O(k^{1.5}) vertex kernel for Connected k-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most graph optimization/decision problems can be solved in polynomial time on graphs of twin-width at most 1.

Cite as

Édouard Bonnet, Eun Jung Kim, Amadeus Reinald, Stéphan Thomassé, and Rémi Watrigant. Twin-Width and Polynomial Kernels. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bonnet_et_al:LIPIcs.IPEC.2021.10,
  author =	{Bonnet, \'{E}douard and Kim, Eun Jung and Reinald, Amadeus and Thomass\'{e}, St\'{e}phan and Watrigant, R\'{e}mi},
  title =	{{Twin-Width and Polynomial Kernels}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-216-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{214},
  editor =	{Golovach, Petr A. and Zehavi, Meirav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.10},
  URN =		{urn:nbn:de:0030-drops-153932},
  doi =		{10.4230/LIPIcs.IPEC.2021.10},
  annote =	{Keywords: Twin-width, kernelization, lower bounds, Dominating Set}
}
Document
Track A: Algorithms, Complexity and Games
4 vs 7 Sparse Undirected Unweighted Diameter is SETH-Hard at Time n^{4/3}

Authors: Édouard Bonnet

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)


Abstract
We show, assuming the Strong Exponential Time Hypothesis, that for every ε > 0, approximating undirected unweighted Diameter on n-vertex m-edge graphs within ratio 7/4 - ε requires m^{4/3 - o(1)} time, even when m = Õ(n). This is the first result that conditionally rules out a near-linear time 5/3-approximation for undirected Diameter.

Cite as

Édouard Bonnet. 4 vs 7 Sparse Undirected Unweighted Diameter is SETH-Hard at Time n^{4/3}. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 34:1-34:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bonnet:LIPIcs.ICALP.2021.34,
  author =	{Bonnet, \'{E}douard},
  title =	{{4 vs 7 Sparse Undirected Unweighted Diameter is SETH-Hard at Time n^\{4/3\}}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.34},
  URN =		{urn:nbn:de:0030-drops-141034},
  doi =		{10.4230/LIPIcs.ICALP.2021.34},
  annote =	{Keywords: Diameter, inapproximability, SETH lower bounds, k-Orthogonal Vectors}
}
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