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**Published in:** LIPIcs, Volume 281, 37th International Symposium on Distributed Computing (DISC 2023)

A proof-labeling scheme (PLS) for a boolean predicate Π on labeled graphs is a mechanism used for certifying the legality with respect to Π of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc.
In 2021, a breakthrough has been obtained, as a "meta-theorem" stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every MSO₂ property Π on labeled graphs, there exists a PLS for Π with O(log n)-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded tree-width, using certificates on O(log² n) bits.
We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every MSO₁ property Π on labeled graphs, there exists a PLS for Π with O(log² n)-bit certificates for all graphs of bounded clique-width. As a consequence, certifying families of graphs such as distance-hereditary graphs and (induced) P₄-free graphs (a.k.a., cographs) can be done using a PLS with O(log² n)-bit certificates, merely because each of these two classes can be specified in MSO₁. In fact, we show that certifying P₄-free graphs can be done with certificates on O(log n) bits only. This is in contrast to the class of C₄-free graphs (which does not have bounded clique-width) which requires Ω̃(√n)-bit certificates.

Pierre Fraigniaud, Frédéric Mazoit, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. Distributed Certification for Classes of Dense Graphs. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fraigniaud_et_al:LIPIcs.DISC.2023.20, author = {Fraigniaud, Pierre and Mazoit, Fr\'{e}d\'{e}ric and Montealegre, Pedro and Rapaport, Ivan and Todinca, Ioan}, title = {{Distributed Certification for Classes of Dense Graphs}}, booktitle = {37th International Symposium on Distributed Computing (DISC 2023)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-301-0}, ISSN = {1868-8969}, year = {2023}, volume = {281}, editor = {Oshman, Rotem}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2023.20}, URN = {urn:nbn:de:0030-drops-191467}, doi = {10.4230/LIPIcs.DISC.2023.20}, annote = {Keywords: CONGEST, Proof Labelling Schemes, clique-width, MSO} }

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**Published in:** LIPIcs, Volume 253, 26th International Conference on Principles of Distributed Systems (OPODIS 2022)

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable. The separation results are obtained by approaching communication complexity through an original angle, which may be of an independent interest. The two players are not bounded to compute the value of a binary function, but the combined outputs of the two players are constrained by this value. In particular, we introduce the XOR-Index problem, in which Alice is given a binary vector x ∈ {0,1}ⁿ together with an index i ∈ [n], Bob is given a binary vector y ∈ {0,1}ⁿ together with an index j ∈ [n], and, after a single round of 2-way communication, Alice must output a boolean out_A, and Bob must output a boolean out_B, such that out_A ∧ out_B = x_j⊕ y_i. We show that the communication complexity of XOR-Index is Ω(n) bits.

Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Ivan Rapaport, Martín Ríos-Wilson, and Ioan Todinca. Computing Power of Hybrid Models in Synchronous Networks. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fraigniaud_et_al:LIPIcs.OPODIS.2022.20, author = {Fraigniaud, Pierre and Montealegre, Pedro and Paredes, Pablo and Rapaport, Ivan and R{\'\i}os-Wilson, Mart{\'\i}n and Todinca, Ioan}, title = {{Computing Power of Hybrid Models in Synchronous Networks}}, booktitle = {26th International Conference on Principles of Distributed Systems (OPODIS 2022)}, pages = {20:1--20:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-265-5}, ISSN = {1868-8969}, year = {2023}, volume = {253}, editor = {Hillel, Eshcar and Palmieri, Roberto and Rivi\`{e}re, Etienne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2022.20}, URN = {urn:nbn:de:0030-drops-176401}, doi = {10.4230/LIPIcs.OPODIS.2022.20}, annote = {Keywords: hybrid model, synchronous networks, LOCAL, CONGEST, Broadcast Congested Clique} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

We show that if the edges or vertices of an undirected graph G can be covered by k shortest paths, then the pathwidth of G is upper-bounded by a function of k. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph G and a set of k pairs of vertices called terminals, asks whether G can be covered by k shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph G and a set of k terminals, asks whether there exist binom(k,2) shortest paths, each joining a distinct pair of terminals such that these paths cover G). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter k.

Maël Dumas, Florent Foucaud, Anthony Perez, and Ioan Todinca. On Graphs Coverable by k Shortest Paths. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dumas_et_al:LIPIcs.ISAAC.2022.40, author = {Dumas, Ma\"{e}l and Foucaud, Florent and Perez, Anthony and Todinca, Ioan}, title = {{On Graphs Coverable by k Shortest Paths}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.40}, URN = {urn:nbn:de:0030-drops-173251}, doi = {10.4230/LIPIcs.ISAAC.2022.40}, annote = {Keywords: Shortest paths, covering problems, parameterized complexity} }

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Brief Announcement

**Published in:** LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable.

Pierre Fraigniaud, Pedro Montealegre, Pablo Paredes, Ivan Rapaport, Martín Ríos-Wilson, and Ioan Todinca. Brief Announcement: Computing Power of Hybrid Models in Synchronous Networks. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 43:1-43:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fraigniaud_et_al:LIPIcs.DISC.2022.43, author = {Fraigniaud, Pierre and Montealegre, Pedro and Paredes, Pablo and Rapaport, Ivan and R{\'\i}os-Wilson, Mart{\'\i}n and Todinca, Ioan}, title = {{Brief Announcement: Computing Power of Hybrid Models in Synchronous Networks}}, booktitle = {36th International Symposium on Distributed Computing (DISC 2022)}, pages = {43:1--43:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-255-6}, ISSN = {1868-8969}, year = {2022}, volume = {246}, editor = {Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.43}, URN = {urn:nbn:de:0030-drops-172345}, doi = {10.4230/LIPIcs.DISC.2022.43}, annote = {Keywords: hybrid model, synchronous networks, LOCAL, CONGEST, Broadcast Congested Clique} }

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

We consider the Strictly Chordal Editing problem, where one is given an undirected graph G = (V,E) and a parameter k ∈ ℕ and seeks to edit (add or delete) at most k edges from G to obtain a strictly chordal graph. Problems Strictly Chordal Completion and Strictly Chordal Deletion are defined similarly, by only allowing edge additions for the former, and only edge deletions for the latter. Strictly chordal graphs are a generalization of 3-leaf power graphs and a subclass of 4-leaf power graphs. They can be defined, e.g., as dart and gem-free chordal graphs. We prove the NP-completeness for all three variants of the problem and provide an O(k³) vertex-kernel for the completion version and O(k⁴) vertex-kernels for the two others.

Maël Dumas, Anthony Perez, and Ioan Todinca. Polynomial Kernels for Strictly Chordal Edge Modification Problems. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dumas_et_al:LIPIcs.IPEC.2021.17, author = {Dumas, Ma\"{e}l and Perez, Anthony and Todinca, Ioan}, title = {{Polynomial Kernels for Strictly Chordal Edge Modification Problems}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {17:1--17:16}, 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.2021.17}, URN = {urn:nbn:de:0030-drops-154005}, doi = {10.4230/LIPIcs.IPEC.2021.17}, annote = {Keywords: Parameterized complexity, kernelization algorithms, graph modification, strictly chordal graphs} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We consider the Trivially Perfect Editing problem, where one is given an undirected graph G = (V,E) and a parameter k ∈ ℕ and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-complete [Burzyn et al., 2006; James Nastos and Yong Gao, 2013] and to admit so-called polynomial kernels [Pål Grønås Drange and Michał Pilipczuk, 2018; Jiong Guo, 2007]. More precisely, the existence of an O(k³) vertex-kernel for Trivially Perfect Completion was announced by Guo [Jiong Guo, 2007] but without a stand-alone proof. More recently, Drange and Pilipczuk [Pål Grønås Drange and Michał Pilipczuk, 2018] provided O(k⁷) vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem.

Maël Dumas, Anthony Perez, and Ioan Todinca. A Cubic Vertex-Kernel for Trivially Perfect Editing. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 45:1-45:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dumas_et_al:LIPIcs.MFCS.2021.45, author = {Dumas, Ma\"{e}l and Perez, Anthony and Todinca, Ioan}, title = {{A Cubic Vertex-Kernel for Trivially Perfect Editing}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {45:1--45:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.45}, URN = {urn:nbn:de:0030-drops-144851}, doi = {10.4230/LIPIcs.MFCS.2021.45}, annote = {Keywords: Parameterized complexity, kernelization algorithms, graph modification, trivially perfect graphs} }

Document

**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

In this paper we present distributed property-testing algorithms for graph properties in the CONGEST model, with emphasis on testing subgraph-freeness. Testing a graph property P means distinguishing graphs G = (V,E) having property P from graphs that are epsilon-far from having it, meaning that epsilon|E| edges must be added or removed from G to obtain a graph satisfying P.
We present a series of results, including:
- Testing H-freeness in O(1/epsilon) rounds, for any constant-sized graph H containing an edge (u,v) such that any cycle in H contain either u or v (or both). This includes all connected graphs over five vertices except K_5. For triangles, we can do even better when epsilon is not too small.
- A deterministic CONGEST protocol determining whether a graph contains a given tree as a subgraph in constant time.
- For cliques K_s with s >= 5, we show that K_s-freeness can be tested in O(m^(1/2-1/(s-2)) epsilon^(-1/2-1/(s-2))) rounds, where m is the number of edges in the network graph.
- We describe a general procedure for converting epsilon-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((log n)/epsilon)+f((log n)/epsilon) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an epsilon-tester for testing whether a graph is bipartite and testing whether a graph is cycle-free. Moreover, for cycle-freeness, we obtain a corrector of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property.
These protocols extend and improve previous results of [Censor-Hillel et al. 2016] and [Fraigniaud et al. 2016].

Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three Notes on Distributed Property Testing. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 15:1-15:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{even_et_al:LIPIcs.DISC.2017.15, author = {Even, Guy and Fischer, Orr and Fraigniaud, Pierre and Gonen, Tzlil and Levi, Reut and Medina, Moti and Montealegre, Pedro and Olivetti, Dennis and Oshman, Rotem and Rapaport, Ivan and Todinca, Ioan}, title = {{Three Notes on Distributed Property Testing}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {15:1--15:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.15}, URN = {urn:nbn:de:0030-drops-79847}, doi = {10.4230/LIPIcs.DISC.2017.15}, annote = {Keywords: Property testing, Property correcting, Distributed algorithms, CONGEST model} }

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