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

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.

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} }

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

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.

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} }

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Track A: Algorithms, Complexity and Games

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

We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time f(d,k)n for n-vertex graphs given with a witness that the twin-width is at most d, called d-contraction sequence or d-sequence, and formulas of size k [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that f is a tower of exponentials of height roughly k. In this paper, we show that algorithms based on twin-width need not be impractical. We present 2^{O(k)}n-time algorithms for k-Independent Set, r-Scattered Set, k-Clique, and k-Dominating Set when an O(1)-sequence of the graph is given in input. We further show how to solve the weighted version of k-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in the slightly worse running time 2^{O(k log k)}n. Up to logarithmic factors in the exponent, all these running times are optimal, unless the Exponential Time Hypothesis fails. Like our FO model checking algorithm, these new algorithms are based on a dynamic programming scheme following the sequence of contractions forward.
We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of such a reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, thereby establishing that bounded twin-width classes are χ-bounded. This significantly extends the χ-boundedness of bounded rank-width classes, and does so with a very concise proof. It readily yields a constant approximation for Max Independent Set on K_t-free graphs of bounded twin-width, and a 2^{O(OPT)}-approximation for Min Coloring on bounded twin-width graphs. We further observe that a constant approximation for Max Independent Set on bounded twin-width graphs (but arbitrarily large clique number) would actually imply a PTAS.
The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. This property is trivially shared with graphs of bounded average degree. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in time O(n log n) and time O(n² log n), respectively. In sharp contrast, even Diameter does not admit a truly subquadratic algorithm on bounded twin-width graphs, unless the Strong Exponential Time Hypothesis fails.
The fourth algorithmic use of twin-width builds on the so-called versatile tree of contractions [Bonnet et al., SODA '21], a branching and more robust witness of low twin-width. We present constant-approximation algorithms for Min Dominating Set and related problems, on bounded twin-width graphs, by showing that the integrality gap is constant. This is done by going down the versatile tree and stopping accordingly to a problem-dependent criterion. At the reached node, a greedy approach yields the desired approximation.

Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: Max Independent Set, Min Dominating Set, and Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bonnet_et_al:LIPIcs.ICALP.2021.35, author = {Bonnet, \'{E}douard and Geniet, Colin and Kim, Eun Jung and Thomass\'{e}, St\'{e}phan and Watrigant, R\'{e}mi}, title = {{Twin-width III: Max Independent Set, Min Dominating Set, and Coloring}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {35:1--35:20}, 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.2021.35}, URN = {urn:nbn:de:0030-drops-141044}, doi = {10.4230/LIPIcs.ICALP.2021.35}, annote = {Keywords: Twin-width, Max Independent Set, Min Dominating Set, Coloring, Parameterized Algorithms, Approximation Algorithms, Exact Algorithms} }

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**Published in:** LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Big step normalisation is a normalisation method for typed lambda-calculi which relies on a purely syntactic recursive evaluator. Termination of that evaluator is proven using a predicate called strong computability, similar to the techniques used to prove strong normalisation of β-reduction for typed lambda-calculi. We generalise big step normalisation to a minimalist dependent type theory. Compared to previous presentations of big step normalisation for e.g. the simply-typed lambda-calculus, we use a quotiented syntax of type theory, which crucially reduces the syntactic complexity introduced by dependent types. Most of the proof has been formalised using Agda.

Thorsten Altenkirch and Colin Geniet. Big Step Normalisation for Type Theory. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{altenkirch_et_al:LIPIcs.TYPES.2019.4, author = {Altenkirch, Thorsten and Geniet, Colin}, title = {{Big Step Normalisation for Type Theory}}, booktitle = {25th International Conference on Types for Proofs and Programs (TYPES 2019)}, pages = {4:1--4:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-158-0}, ISSN = {1868-8969}, year = {2020}, volume = {175}, editor = {Bezem, Marc and Mahboubi, Assia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.4}, URN = {urn:nbn:de:0030-drops-130682}, doi = {10.4230/LIPIcs.TYPES.2019.4}, annote = {Keywords: Normalisation, big step normalisation, type theory, dependent types, Agda} }

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