21 Search Results for "Torunczyk, Szymon"


Document
Invited Talk
Structurally Tractable Graph Classes (Invited Talk)

Authors: Szymon Toruńczyk

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
Sparsity theory, initiated by Ossona de Mendez and Nešetřil, identifies those classes of sparse graphs that are tractable in various ways - algorithmically, combinatorially, and logically - as exactly the nowhere dense classes. An ongoing effort aims at generalizing sparsity theory to classes of graphs that are not necessarily sparse. Twin-width theory, developed by Bonnet, Thomassé and co-authors, is a step in that direction. A theory unifying the two is anticipated. It is conjectured that the relevant notion characterising dense graph classes that are tractable, generalising nowhere denseness and bounded twin-width, is the notion of a monadically dependent class, introduced by Shelah in model theory. I will survey the recent, rapid progress in the understanding of those classes, and of the related monadically stable classes. This development combines tools from structural graph theory, logic (finite and infinite model theory), and algorithms (parameterised algorithms and range search queries).

Cite as

Szymon Toruńczyk. Structurally Tractable Graph Classes (Invited Talk). In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{torunczyk:LIPIcs.STACS.2024.3,
  author =	{Toru\'{n}czyk, Szymon},
  title =	{{Structurally Tractable Graph Classes}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.3},
  URN =		{urn:nbn:de:0030-drops-197134},
  doi =		{10.4230/LIPIcs.STACS.2024.3},
  annote =	{Keywords: Structural graph theory, Monadic dependence, monadic NIP, twin-width}
}
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-dev.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-dev.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
Track B: Automata, Logic, Semantics, and Theory of Programming
Flipper Games for Monadically Stable Graph Classes

Authors: Jakub Gajarský, Nikolas Mählmann, Rose McCarty, Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, Sebastian Siebertz, Marek Sokołowski, and Szymon Toruńczyk

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


Abstract
A class of graphs C is monadically stable if for every unary expansion Ĉ of C, one cannot encode - using first-order transductions - arbitrarily long linear orders in graphs from C. It is known that nowhere dense graph classes are monadically stable; these include classes of bounded maximum degree and classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms that is also suited for capturing structure in dense graphs. In this work we provide a characterization of monadic stability in terms of the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Localizer, who in each round is forced to restrict the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J. ACM '17). We give two different proofs of our main result. The first proof is based on tools borrowed from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we show that monadic stability for graph classes coincides with monadic stability of existential formulas with two free variables, and we provide another combinatorial characterization of monadic stability via forbidden patterns. The second proof relies on the recently introduced notion of flip-flatness (Dreier, Mählmann, Siebertz, and Toruńczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper’s moves in a winning strategy.

Cite as

Jakub Gajarský, Nikolas Mählmann, Rose McCarty, Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, Sebastian Siebertz, Marek Sokołowski, and Szymon Toruńczyk. Flipper Games for Monadically Stable Graph Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 128:1-128:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2023.128,
  author =	{Gajarsk\'{y}, Jakub and M\"{a}hlmann, Nikolas and McCarty, Rose and Ohlmann, Pierre and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Siebertz, Sebastian and Soko{\l}owski, Marek and Toru\'{n}czyk, Szymon},
  title =	{{Flipper Games for Monadically Stable Graph Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{128:1--128:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.128},
  URN =		{urn:nbn:de:0030-drops-181804},
  doi =		{10.4230/LIPIcs.ICALP.2023.128},
  annote =	{Keywords: Stability theory, structural graph theory, games}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes

Authors: Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk

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


Abstract
We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class 𝒞, and using a first-order formula φ with parameters we are able to define, in every graph G ∈ 𝒞, a relation R that satisfies some hereditary first-order assertion ψ. Then we are able to find a first-order formula φ' that has the same property, but additionally is finitary: there is finite bound k ∈ ℕ such that in every graph G ∈ 𝒞, different choices of parameters give only at most k different relations R that can be defined using φ'. We use the Finitary Substitute Lemma to derive two corollaries about the existence of certain canonical decompositions in classes of well-structured graphs. - We prove that in the Splitter game, which characterizes nowhere dense graph classes, and in the Flipper game, which characterizes monadically stable graph classes, there is a winning strategy for Splitter, respectively Flipper, that can be defined in first-order logic from the game history. Thus, the strategy is canonical. - We show that for any fixed graph class 𝒞 of bounded shrubdepth, there is an 𝒪(n²)-time algorithm that given an n-vertex graph G ∈ 𝒞, computes in an isomorphism-invariant way a structure H of bounded treedepth in which G can be interpreted. A corollary of this result is an 𝒪(n²)-time isomorphism test and canonization algorithm for any fixed class of bounded shrubdepth.

Cite as

Pierre Ohlmann, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk. Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 135:1-135:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{ohlmann_et_al:LIPIcs.ICALP.2023.135,
  author =	{Ohlmann, Pierre and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Toru\'{n}czyk, Szymon},
  title =	{{Canonical Decompositions in Monadically Stable and Bounded Shrubdepth Graph Classes}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{135:1--135:17},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.135},
  URN =		{urn:nbn:de:0030-drops-181874},
  doi =		{10.4230/LIPIcs.ICALP.2023.135},
  annote =	{Keywords: Model Theory, Stability Theory, Shrubdepth, Nowhere Dense, Monadically Stable}
}
Document
On Rational Recursive Sequences

Authors: Lorenzo Clemente, Maria Donten-Bury, Filip Mazowiecki, and Michał Pilipczuk

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


Abstract
We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational functions of the previous values. An alternative class is that of simple ratrec sequences, where one uses a single recursive equation, however of depth k: the next value is defined as a rational function of k previous values. We conjecture that the classes ratrec and simple ratrec coincide. The main contribution of this paper is a proof of a variant of this conjecture where the initial conditions are treated symbolically, using a formal variable per sequence, while the sequences themselves consist of rational functions over those variables. While the initial conjecture does not follow from this variant, we hope that the introduced algebraic techniques may eventually be helpful in resolving the problem. The class ratrec strictly generalises a well-known class of polynomial recursive sequences (polyrec). These are defined like ratrec, but using polynomial functions instead of rational ones. One can observe that if our conjecture is true and effective, then we can improve the complexities of the zeroness and the equivalence problems for polyrec sequences. Currently, the only known upper bound is Ackermanian, which follows from results on polynomial automata. We complement this observation by proving a PSPACE lower bound for both problems for polyrec. Our lower bound construction also implies that the Skolem problem is PSPACE-hard for the polyrec class.

Cite as

Lorenzo Clemente, Maria Donten-Bury, Filip Mazowiecki, and Michał Pilipczuk. On Rational Recursive Sequences. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 24:1-24:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{clemente_et_al:LIPIcs.STACS.2023.24,
  author =	{Clemente, Lorenzo and Donten-Bury, Maria and Mazowiecki, Filip and Pilipczuk, Micha{\l}},
  title =	{{On Rational Recursive Sequences}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{24:1--24:21},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.24},
  URN =		{urn:nbn:de:0030-drops-176763},
  doi =		{10.4230/LIPIcs.STACS.2023.24},
  annote =	{Keywords: recursive sequences, polynomial automata, zeroness problem, equivalence problem}
}
Document
Track A: Algorithms, Complexity and Games
Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures

Authors: Michał Pilipczuk, Nicole Schirrmacher, Sebastian Siebertz, Szymon Toruńczyk, and Alexandre Vigny

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


Abstract
We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer parameter k, we can in fixed-parameter time construct a data structure that can later be used to answer queries of the form: "are vertices s and t connected via a path that avoids vertices u₁,…, u_k?" in time 2^𝒪(k). In the terminology of the literature on data structures, this gives the first deterministic data structure for connectivity under vertex failures where for every fixed number of failures, all operations can be performed in constant time. With the aim to understand the power and the limitations of our new techniques, we prove an algorithmic meta theorem for the recently introduced separator logic, which extends first-order logic with atoms for connectivity under vertex failures. We prove that the model-checking problem for separator logic is fixed-parameter tractable on every class of graphs that exclude a fixed topological minor. We also show a weak converse. This implies that from the point of view of parameterized complexity, under standard complexity theoretical assumptions, the frontier of tractability of separator logic is almost exactly delimited by classes excluding a fixed topological minor. The backbone of our proof relies on a decomposition theorem of Cygan, Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh [SICOMP '19], which provides a tree decomposition of a given graph into bags that are unbreakable. Crucially, unbreakability allows to reduce separator logic to plain first-order logic within each bag individually. Guided by this observation, we design our model-checking algorithm using dynamic programming over the tree decomposition, where the transition at each bag amounts to running a suitable model-checking subprocedure for plain first-order logic. This approach is robust enough to provide also an extension to efficient enumeration of answers to a query expressed in separator logic.

Cite as

Michał Pilipczuk, Nicole Schirrmacher, Sebastian Siebertz, Szymon Toruńczyk, and Alexandre Vigny. Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 102:1-102:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{pilipczuk_et_al:LIPIcs.ICALP.2022.102,
  author =	{Pilipczuk, Micha{\l} and Schirrmacher, Nicole and Siebertz, Sebastian and Toru\'{n}czyk, Szymon and Vigny, Alexandre},
  title =	{{Algorithms and Data Structures for First-Order Logic with Connectivity Under Vertex Failures}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{102:1--102:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.102},
  URN =		{urn:nbn:de:0030-drops-164432},
  doi =		{10.4230/LIPIcs.ICALP.2022.102},
  annote =	{Keywords: Combinatorics and graph theory, Computational applications of logic, Data structures, Fixed-parameter algorithms and complexity, Graph algorithms}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
Twin-Width and Types

Authors: Jakub Gajarský, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk

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


Abstract
We study problems connected to first-order logic in graphs of bounded twin-width. Inspired by the approach of Bonnet et al. [FOCS 2020], we introduce a robust methodology of local types and describe their behavior in contraction sequences - the decomposition notion underlying twin-width. We showcase the applicability of the methodology by proving the following two algorithmic results. In both statements, we fix a first-order formula φ(x_1,…,x_k) and a constant d, and we assume that on input we are given a graph G together with a contraction sequence of width at most d. - One can in time 𝒪(n) construct a data structure that can answer the following queries in time 𝒪(log log n): given w_1,…,w_k, decide whether φ(w_1,…,w_k) holds in G. - After 𝒪(n)-time preprocessing, one can enumerate all tuples w₁,…,w_k that satisfy φ(x_1,…,x_k) in G with 𝒪(1) delay. In the first case, the query time can be reduced to 𝒪(1/ε) at the expense of increasing the construction time to 𝒪(n^{1+ε}), for any fixed ε > 0. Finally, we also apply our tools to prove the following statement, which shows optimal bounds on the VC density of set systems that are first-order definable in graphs of bounded twin-width. - Let G be a graph of twin-width d, A be a subset of vertices of G, and φ(x_1,…,x_k,y_1,…,y_l) be a first-order formula. Then the number of different subsets of A^k definable by φ using l-tuples of vertices from G as parameters, is bounded by O(|A|^l).

Cite as

Jakub Gajarský, Michał Pilipczuk, Wojciech Przybyszewski, and Szymon Toruńczyk. Twin-Width and Types. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 123:1-123:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2022.123,
  author =	{Gajarsk\'{y}, Jakub and Pilipczuk, Micha{\l} and Przybyszewski, Wojciech and Toru\'{n}czyk, Szymon},
  title =	{{Twin-Width and Types}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{123:1--123:21},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.123},
  URN =		{urn:nbn:de:0030-drops-164640},
  doi =		{10.4230/LIPIcs.ICALP.2022.123},
  annote =	{Keywords: twin-width, FO logic, model checking, query answering, enumeration}
}
Document
First-Order Logic with Connectivity Operators

Authors: Nicole Schirrmacher, Sebastian Siebertz, and Alexandre Vigny

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)


Abstract
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates conn_k(x,y,z_1,…,z_k) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z_1,…,z_k have been deleted, we obtain separator logic FO+conn. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO+conn_k the fragment of separator logic that is restricted to connectivity predicates with at most k+2 variables (that is, at most k deletions). We show that FO+conn_{k+1} is strictly more expressive than FO+conn_k for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO+DP by adding the atomic predicates disjoint-paths_k[(x_1,y_1),…,(x_k,y_k)] that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) x_i and y_i for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again we show that the fragments FO+DP_k that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

Cite as

Nicole Schirrmacher, Sebastian Siebertz, and Alexandre Vigny. First-Order Logic with Connectivity Operators. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{schirrmacher_et_al:LIPIcs.CSL.2022.34,
  author =	{Schirrmacher, Nicole and Siebertz, Sebastian and Vigny, Alexandre},
  title =	{{First-Order Logic with Connectivity Operators}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.34},
  URN =		{urn:nbn:de:0030-drops-157548},
  doi =		{10.4230/LIPIcs.CSL.2022.34},
  annote =	{Keywords: First-order logic, graph theory, connectivity}
}
Document
Progressive Algorithms for Domination and Independence

Authors: Grzegorz Fabiański, Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)


Abstract
We consider a generic algorithmic paradigm that we call progressive exploration, which can be used to develop simple and efficient parameterized graph algorithms. We identify two model-theoretic properties that lead to efficient progressive algorithms, namely variants of the Helly property and stability. We demonstrate our approach by giving linear-time fixed-parameter algorithms for the Distance-r Dominating Set problem (parameterized by the solution size) in a wide variety of restricted graph classes, such as powers of nowhere dense classes, map graphs, and (for r=1) biclique-free graphs. Similarly, for the Distance-r Independent Set problem the technique can be used to give a linear-time fixed-parameter algorithm on any nowhere dense class. Despite the simplicity of the method, in several cases our results extend known boundaries of tractability for the considered problems and improve the best known running times.

Cite as

Grzegorz Fabiański, Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. Progressive Algorithms for Domination and Independence. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{fabianski_et_al:LIPIcs.STACS.2019.27,
  author =	{Fabia\'{n}ski, Grzegorz and Pilipczuk, Micha{\l} and Siebertz, Sebastian and Toru\'{n}czyk, Szymon},
  title =	{{Progressive Algorithms for Domination and Independence}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{27:1--27:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.27},
  URN =		{urn:nbn:de:0030-drops-102660},
  doi =		{10.4230/LIPIcs.STACS.2019.27},
  annote =	{Keywords: Dominating Set, Independent Set, nowhere denseness, stability, fixed-parameter tractability}
}
Document
First-Order Interpretations of Bounded Expansion Classes

Authors: Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)


Abstract
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

Cite as

Jakub Gajarský, Stephan Kreutzer, Jaroslav Nesetril, Patrice Ossona de Mendez, Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. First-Order Interpretations of Bounded Expansion Classes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 126:1-126:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{gajarsky_et_al:LIPIcs.ICALP.2018.126,
  author =	{Gajarsk\'{y}, Jakub and Kreutzer, Stephan and Nesetril, Jaroslav and Ossona de Mendez, Patrice and Pilipczuk, Michal and Siebertz, Sebastian and Torunczyk, Szymon},
  title =	{{First-Order Interpretations of Bounded Expansion Classes}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{126:1--126:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.126},
  URN =		{urn:nbn:de:0030-drops-91300},
  doi =		{10.4230/LIPIcs.ICALP.2018.126},
  annote =	{Keywords: Logical interpretations/transductions, structurally sparse graphs, bounded expansion}
}
Document
Entropy Bounds for Conjunctive Queries with Functional Dependencies

Authors: Tomasz Gogacz and Szymon Torunczyk

Published in: LIPIcs, Volume 68, 20th International Conference on Database Theory (ICDT 2017)


Abstract
We study the problem of finding the worst-case size of the result Q(D) of a fixed conjunctive query Q applied to a database D satisfying given functional dependencies. We provide a characterization of this bound in terms of entropy vectors, and in terms of finite groups. In particular, we show that an upper bound provided by [Gottlob, Lee, Valiant and Valiant, J.ACM, 2012] is tight, and that a correspondence of [Chan and Yeung, ACM TOIT, 2002] is preserved in the presence of functional dependencies. However, tightness of a weaker upper bound provided by Gottlob et al., which would have immediate applications to evaluation of join queries ([Khamis, Ngo, and Suciu, PODS, 2016]) remains open. Our result shows that the problem of computing the worst-case size bound, in the general case, is closely related to difficult problems from information theory.

Cite as

Tomasz Gogacz and Szymon Torunczyk. Entropy Bounds for Conjunctive Queries with Functional Dependencies. In 20th International Conference on Database Theory (ICDT 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 68, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{gogacz_et_al:LIPIcs.ICDT.2017.15,
  author =	{Gogacz, Tomasz and Torunczyk, Szymon},
  title =	{{Entropy Bounds for Conjunctive Queries with Functional Dependencies}},
  booktitle =	{20th International Conference on Database Theory (ICDT 2017)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-024-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{68},
  editor =	{Benedikt, Michael and Orsi, Giorgio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2017.15},
  URN =		{urn:nbn:de:0030-drops-70479},
  doi =		{10.4230/LIPIcs.ICDT.2017.15},
  annote =	{Keywords: database theory, conjunctive queries, size bounds, entropy, finite groups, entropy cone}
}
Document
Homomorphism Problems for First-Order Definable Structures

Authors: Bartek Klin, Slawomir Lasota, Joanna Ochremiak, and Szymon Torunczyk

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)


Abstract
We investigate several variants of the homomorphism problem: given two relational structures, is there a homomorphism from one to the other? The input structures are possibly infinite, but definable by first-order interpretations in a fixed structure. Their signatures can be either finite or infinite but definable. The homomorphisms can be either arbitrary, or definable with parameters, or definable without parameters. For each of these variants, we determine its decidability status.

Cite as

Bartek Klin, Slawomir Lasota, Joanna Ochremiak, and Szymon Torunczyk. Homomorphism Problems for First-Order Definable Structures. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{klin_et_al:LIPIcs.FSTTCS.2016.14,
  author =	{Klin, Bartek and Lasota, Slawomir and Ochremiak, Joanna and Torunczyk, Szymon},
  title =	{{Homomorphism Problems for First-Order Definable Structures}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.14},
  URN =		{urn:nbn:de:0030-drops-68498},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.14},
  annote =	{Keywords: Sets with atoms, first-order interpretations, homomorphism problem}
}
Document
Models of Lambda-Calculus and the Weak MSO Logic

Authors: Pawel Parys and Szymon Torunczyk

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)


Abstract
We study the weak MSO logic in relationship to infinitary lambda-calculus. We show that for every formula phi of weak MSO there exists a finitary model of infinitary lambda-calculus recognizing the set of infinitary lambda-terms whose Böhm tree satisfies phi. The model is effective, in the sense that for every lambda-Y-term we can effectively compute its value in the model. In particular, given a finite lambda-Y-term, one can decide whether the resulting Böhm tree satisfies a given formula of weak MSO, which is a special case of the result of Ong, which concerns unrestricted MSO. The existence of effective models for weak MSO and MSO was proved earlier by Salvati and Walukiewicz but our proof uses a different method, as it does not involve automata, but works directly with logics.

Cite as

Pawel Parys and Szymon Torunczyk. Models of Lambda-Calculus and the Weak MSO Logic. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 11:1-11:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{parys_et_al:LIPIcs.CSL.2016.11,
  author =	{Parys, Pawel and Torunczyk, Szymon},
  title =	{{Models of Lambda-Calculus and the Weak MSO Logic}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{11:1--11:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.11},
  URN =		{urn:nbn:de:0030-drops-65511},
  doi =		{10.4230/LIPIcs.CSL.2016.11},
  annote =	{Keywords: typed lambda-calculus, models, weak MSO logic}
}
Document
Non-Homogenizable Classes of Finite Structures

Authors: Albert Atserias and Szymon Torunczyk

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)


Abstract
Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability.

Cite as

Albert Atserias and Szymon Torunczyk. Non-Homogenizable Classes of Finite Structures. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{atserias_et_al:LIPIcs.CSL.2016.16,
  author =	{Atserias, Albert and Torunczyk, Szymon},
  title =	{{Non-Homogenizable Classes of Finite Structures}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.16},
  URN =		{urn:nbn:de:0030-drops-65563},
  doi =		{10.4230/LIPIcs.CSL.2016.16},
  annote =	{Keywords: Fra\"{i}ss\'{e} class, amalgmation class, reduct, Constraint Satisfaction Problem, bounded width}
}
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