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Documents authored by Sokołowski, Marek

Document
DOI: 10.4230/LIPIcs.ISAAC.2023.11
Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width

Authors: Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
Twin-width is a structural width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual reduction (a contraction sequence) of the given graph down to a single vertex while maintaining limited difference of neighbourhoods of the vertices, and it can be seen as widely generalizing several other traditional structural parameters. Having such a sequence at hand allows to solve many otherwise hard problems efficiently. Our paper focuses on a comparison of twin-width to the more traditional tree-width on sparse graphs. Namely, we prove that if a graph G of twin-width at most 2 contains no K_{t,t} subgraph for some integer t, then the tree-width of G is bounded by a polynomial function of t. As a consequence, for any sparse graph class C we obtain a polynomial time algorithm which for any input graph G ∈ C either outputs a contraction sequence of width at most c (where c depends only on C), or correctly outputs that G has twin-width more than 2. On the other hand, we present an easy example of a graph class of twin-width 3 with unbounded tree-width, showing that our result cannot be extended to higher values of twin-width.
Cite as

Benjamin Bergougnoux, Jakub Gajarský, Grzegorz Guśpiel, Petr Hliněný, Filip Pokrývka, and Marek Sokołowski. Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bergougnoux_et_al:LIPIcs.ISAAC.2023.11,
  author =	{Bergougnoux, Benjamin and Gajarsk\'{y}, Jakub and Gu\'{s}piel, Grzegorz and Hlin\v{e}n\'{y}, Petr and Pokr\'{y}vka, Filip and Soko{\l}owski, Marek},
  title =	{{Sparse Graphs of Twin-Width 2 Have Bounded Tree-Width}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{11:1--11:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.11},
  URN =		{urn:nbn:de:0030-drops-193130},
  doi =		{10.4230/LIPIcs.ISAAC.2023.11},
  annote =	{Keywords: twin-width, tree-width, excluded grid, sparsity}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.128
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.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
DOI: 10.4230/LIPIcs.STACS.2023.46
Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number

Authors: Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski

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

Abstract
Let 𝜑 be a sentence of CMSO₂ (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether 𝜑 is satisfied in G. The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time O_{𝜑,k}(log n). By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time O_k(log n). Our data structure also works in a larger generality of relational structures over binary signatures.
Cite as

Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski. Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{majewski_et_al:LIPIcs.STACS.2023.46,
  author =	{Majewski, Konrad and Pilipczuk, Micha{\l} and Soko{\l}owski, Marek},
  title =	{{Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{46:1--46:13},
  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.2023.46},
  URN =		{urn:nbn:de:0030-drops-176981},
  doi =		{10.4230/LIPIcs.STACS.2023.46},
  annote =	{Keywords: feedback vertex set, CMSO₂ formula, data structure, dynamic graphs, fixed-parameter tractability}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.93
Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás' Path Argument

Authors: Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, and Marek Sokołowski

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

Abstract
We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S_{t,t,t} as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time 2^𝒪(√nlog n) and a quasipolynomial-time approximation scheme with improved running time 2^𝒪(ε^{-1} log⁵ n). The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in P_t-free graphs, ensures that given an n-vertex P_t-free graph, in polynomial time we can find a set P of at most t-1 vertices, such that every connected component of G-N[P] has at most n/2 vertices. Our main technical contribution is an analog of this result for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, in polynomial time we can find a set P of 𝒪(t log n) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n/2 vertices.
Cite as

Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, and Marek Sokołowski. Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás' Path Argument. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{majewski_et_al:LIPIcs.ICALP.2022.93,
  author =	{Majewski, Konrad and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana and Okrasa, Karolina and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l} and Soko{\l}owski, Marek},
  title =	{{Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gy\'{a}rf\'{a}s' Path Argument}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{93:1--93:19},
  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.2022.93},
  URN =		{urn:nbn:de:0030-drops-164343},
  doi =		{10.4230/LIPIcs.ICALP.2022.93},
  annote =	{Keywords: Max Independent Set, subdivided claw, QPTAS, subexponential-time algorithm}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.52
Compact Representation for Matrices of Bounded Twin-Width

Authors: Michał Pilipczuk, Marek Sokołowski, and Anna Zych-Pawlewicz

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
For every fixed d ∈ ℕ, we design a data structure that represents a binary n × n matrix that is d-twin-ordered. The data structure occupies 𝒪_d(n) bits, which is the least one could hope for, and can be queried for entries of the matrix in time 𝒪_d(log log n) per query.
Cite as

Michał Pilipczuk, Marek Sokołowski, and Anna Zych-Pawlewicz. Compact Representation for Matrices of Bounded Twin-Width. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pilipczuk_et_al:LIPIcs.STACS.2022.52,
  author =	{Pilipczuk, Micha{\l} and Soko{\l}owski, Marek and Zych-Pawlewicz, Anna},
  title =	{{Compact Representation for Matrices of Bounded Twin-Width}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.52},
  URN =		{urn:nbn:de:0030-drops-158620},
  doi =		{10.4230/LIPIcs.STACS.2022.52},
  annote =	{Keywords: twin-width, compact representation, adjacency oracle}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.71
Determining 4-Edge-Connected Components in Linear Time

Authors: Wojciech Nadara, Mateusz Radecki, Marcin Smulewicz, and Marek Sokołowski

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

Abstract
In this work, we present the first linear time deterministic algorithm computing the 4-edge-connected components of an undirected graph. First, we show an algorithm listing all 3-edge-cuts in a given 3-edge-connected graph, and then we use the output of this algorithm in order to determine the 4-edge-connected components of the graph.
Cite as

Wojciech Nadara, Mateusz Radecki, Marcin Smulewicz, and Marek Sokołowski. Determining 4-Edge-Connected Components in Linear Time. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{nadara_et_al:LIPIcs.ESA.2021.71,
  author =	{Nadara, Wojciech and Radecki, Mateusz and Smulewicz, Marcin and Soko{\l}owski, Marek},
  title =	{{Determining 4-Edge-Connected Components in Linear Time}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{71:1--71:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.71},
  URN =		{urn:nbn:de:0030-drops-146523},
  doi =		{10.4230/LIPIcs.ESA.2021.71},
  annote =	{Keywords: graphs, connectivity, cuts}
}
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