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Documents authored by Perez, Anthony

Document
DOI: 10.4230/LIPIcs.IPEC.2025.13
On Maximum 2-Clubs

Authors: Joanne Dumont, Michael Lampis, Mathieu Liedloff, Anthony Perez, and Ioan Todinca

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
We consider the Maximum 2-Club problem where one is given as input an undirected graph G = (V,E) and seeks a subset of vertices S of maximum size such that any pair of vertices in S is connected by a path of length at most 2 in the graph induced by S. This problem is a natural relaxation of the famous Maximum Clique problem where any pair of vertices must be connected by an edge. Maximum 2-Club has been well-studied and is known to be NP-complete even on split graphs. It can be solved exactly in O^*(1.62ⁿ) time, where n denotes the number of vertices of the input graph, while being polynomial-time solvable on several graph classes. Parameterized algorithms for structural parameters have also been considered, leading in particular to an algorithm with a double-exponential dependence in the parameter treewidth. Such an algorithm is actually the best one known for the larger parameter vertex cover size up to a constant in the exponent. We provide new results in both directions. We first prove that the double-exponential dependence for parameter vertex cover size is unavoidable under the Exponential Time Hypothesis (ETH). This answers a question left open by Hartung, Komusiewicz, Nichterlein and Suchỳ [Hartung et al., 2015]. Our result also implies that the problem cannot be solved in time sub-exponential in n even for split graphs. We then provide an exact algorithm for the problem restricted to chordal graphs, running in O^*(1.1996ⁿ) time, by reducing Maximum 2-Club on this class to Maximum Independent Set on arbitrary graphs with the same number of vertices. The same reduction shows that we can enumerate all maximum (and inclusion-wise maximal) 2-clubs of a chordal graph in O^*(3^{n/3}) = O^*(1.4423ⁿ) time. We conclude by providing a construction of split graphs with Ω(3^{n/3}/poly(n)) maximum2-clubs, for some polynomial poly showing that the bound for enumeration is essentially tight.
Cite as

Joanne Dumont, Michael Lampis, Mathieu Liedloff, Anthony Perez, and Ioan Todinca. On Maximum 2-Clubs. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dumont_et_al:LIPIcs.IPEC.2025.13,
  author =	{Dumont, Joanne and Lampis, Michael and Liedloff, Mathieu and Perez, Anthony and Todinca, Ioan},
  title =	{{On Maximum 2-Clubs}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.13},
  URN =		{urn:nbn:de:0030-drops-251454},
  doi =		{10.4230/LIPIcs.IPEC.2025.13},
  annote =	{Keywords: 2-clubs, chordal graphs, SETH, parameterized algorithms}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2023.15
An Improved Kernelization Algorithm for Trivially Perfect Editing

Authors: Maël Dumas and Anthony Perez

Published in: LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Abstract
In the Trivially Perfect Editing problem one is given an undirected graph G = (V,E) and an integer k and seeks to add or delete at most k edges in G to obtain a trivially perfect graph. In a recent work, Dumas et al. [Dumas et al., 2023] proved that this problem admits a kernel with O(k³) vertices. This result heavily relies on the fact that the size of trivially perfect modules can be bounded by O(k²) as shown by Drange and Pilipczuk [Drange and Pilipczuk, 2018]. To obtain their cubic vertex-kernel, Dumas et al. [Dumas et al., 2023] then showed that a more intricate structure, so-called comb, can be reduced to O(k²) vertices. In this work we show that the bound can be improved to O(k) for both aforementioned structures and thus obtain a kernel with O(k²) vertices. Our approach relies on the straightforward yet powerful observation that any large enough structure contains unaffected vertices whose neighborhood remains unchanged by an editing of size k, implying strong structural properties.
Cite as

Maël Dumas and Anthony Perez. An Improved Kernelization Algorithm for Trivially Perfect Editing. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dumas_et_al:LIPIcs.IPEC.2023.15,
  author =	{Dumas, Ma\"{e}l and Perez, Anthony},
  title =	{{An Improved Kernelization Algorithm for Trivially Perfect Editing}},
  booktitle =	{18th International Symposium on Parameterized and Exact Computation (IPEC 2023)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-305-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{285},
  editor =	{Misra, Neeldhara and Wahlstr\"{o}m, Magnus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.15},
  URN =		{urn:nbn:de:0030-drops-194340},
  doi =		{10.4230/LIPIcs.IPEC.2023.15},
  annote =	{Keywords: Parameterized complexity, kernelization algorithms, graph modification, trivially perfect graphs}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.40
On Graphs Coverable by k Shortest Paths

Authors: Maël Dumas, Florent Foucaud, Anthony Perez, and Ioan Todinca

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.17
Polynomial Kernels for Strictly Chordal Edge Modification Problems

Authors: Maël Dumas, Anthony Perez, and Ioan Todinca

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.45
A Cubic Vertex-Kernel for Trivially Perfect Editing

Authors: Maël Dumas, Anthony Perez, and Ioan Todinca

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

Abstract
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.
Cite as

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
DOI: 10.4230/LIPIcs.FSTTCS.2009.2305
Kernels for Feedback Arc Set In Tournaments

Authors: Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé

Published in: LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Abstract
A tournament $T = (V,A)$ is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on $n$ vertices and an integer parameter $k$, the {\sc Feedback Arc Set} problem asks whether thegiven digraph has a set of $k$ arcs whose removal results in an acyclicdigraph. The {\sc Feedback Arc Set} problem restricted to tournaments is knownas the {\sc $k$-Feedback Arc Set in Tournaments ($k$-FAST)} problem. In thispaper we obtain a linear vertex kernel for \FAST{}. That is, we give apolynomial time algorithm which given an input instance $T$ to \FAST{} obtains an equivalent instance $T'$ on $O(k)$ vertices. In fact, given any fixed $\epsilon > 0$, the kernelized instance has at most $(2 + \epsilon)k$ vertices.Our result improves the previous known bound of $O(k^2)$ on the kernel size for\FAST{}. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for \FAST.
Cite as

Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé. Kernels for Feedback Arc Set In Tournaments. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 37-47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bessy_et_al:LIPIcs.FSTTCS.2009.2305,
  author =	{Bessy, St\'{e}phane and Fomin, Fedor V. and Gaspers, Serge and Paul, Christophe and Perez, Anthony and Saurabh, Saket and Thomass\'{e}, St\'{e}phan},
  title =	{{Kernels for Feedback Arc Set In Tournaments}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{37--47},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Kannan, Ravi and Narayan Kumar, K.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2305},
  URN =		{urn:nbn:de:0030-drops-23055},
  doi =		{10.4230/LIPIcs.FSTTCS.2009.2305},
  annote =	{Keywords: Parameterized complexity, kernels, tournaments}
}
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