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

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.

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} }

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

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of additive classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most k, for k ≥ 1 form a strict hierarchy in the FO transduction quasiorder.

Jaroslav Nešetřil, Patrice Ossona de Mendez, and Sebastian Siebertz. Structural Properties of the First-Order Transduction Quasiorder. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{nesetril_et_al:LIPIcs.CSL.2022.31, author = {Ne\v{s}et\v{r}il, Jaroslav and Ossona de Mendez, Patrice and Siebertz, Sebastian}, title = {{Structural Properties of the First-Order Transduction Quasiorder}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {31:1--31:16}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.31}, URN = {urn:nbn:de:0030-drops-157514}, doi = {10.4230/LIPIcs.CSL.2022.31}, annote = {Keywords: Finite model theory, first-order transductions, structural graph theory} }

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Invited Talk

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

This paper is an extended abstract of my STACS 2021 talk "First-order transductions of graphs".

Patrice Ossona de Mendez. First-Order Transductions of Graphs (Invited Talk). In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 2:1-2:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ossonademendez:LIPIcs.STACS.2021.2, author = {Ossona de Mendez, Patrice}, title = {{First-Order Transductions of Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {2:1--2:7}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus 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.2021.2}, URN = {urn:nbn:de:0030-drops-136473}, doi = {10.4230/LIPIcs.STACS.2021.2}, annote = {Keywords: Finite model theory, structural graph theory} }

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

The notions of bounded expansion [Nešetřil and Ossona de Mendez, 2008] and nowhere denseness [Nešetřil and Ossona de Mendez, 2011], introduced by Nešetřil and Ossona de Mendez as structural measures for undirected graphs, have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs, introduced by Kreutzer and Tazari [Kreutzer and Tazari, 2012]. The classes of directed graphs having those properties are very general classes of sparse directed graphs, as they include, on one hand, all classes of directed graphs whose underlying undirected class has bounded expansion, such as planar, bounded-genus, and H-minor-free graphs, and on the other hand, they also contain classes whose underlying undirected class is not even nowhere dense. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion and nowhere crownful classes.

Stephan Kreutzer, Irene Muzi, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Algorithmic Properties of Sparse Digraphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kreutzer_et_al:LIPIcs.STACS.2019.46, author = {Kreutzer, Stephan and Muzi, Irene and Ossona de Mendez, Patrice and Rabinovich, Roman and Siebertz, Sebastian}, title = {{Algorithmic Properties of Sparse Digraphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {46:1--46:20}, 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.2019.46}, URN = {urn:nbn:de:0030-drops-102859}, doi = {10.4230/LIPIcs.STACS.2019.46}, annote = {Keywords: Directed graphs, graph algorithms, parameterized complexity, approximation} }

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

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.

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.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} }

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