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**Published in:** LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

For any set system ℋ = (V,ℛ), ℛ ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in ℛ. The VC-dimension of ℋ is the size of a largest shattered set in V. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph G = (V,E), the VC-dimension of G is defined as the VC-dimension of (V, N), where N contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ V in G. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the W[1]-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.

David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Practical Computation of Graph VC-Dimension. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coudert_et_al:LIPIcs.SEA.2024.8, author = {Coudert, David and Csik\'{o}s, M\'{o}nika and Ducoffe, Guillaume and Viennot, Laurent}, title = {{Practical Computation of Graph VC-Dimension}}, booktitle = {22nd International Symposium on Experimental Algorithms (SEA 2024)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-325-6}, ISSN = {1868-8969}, year = {2024}, volume = {301}, editor = {Liberti, Leo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.8}, URN = {urn:nbn:de:0030-drops-203731}, doi = {10.4230/LIPIcs.SEA.2024.8}, annote = {Keywords: VC-dimension, graph, algorithm} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let Ext_α be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than α pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every m-edge graph in Ext_{α} can be computed in deterministic O(α³ m^{3/2}) time. We then improve the runtime to O(α³m) for bipartite graphs, to O(α⁵m) for triangle-free graphs, O(α³Δm) for graphs with maximum degree Δ, and more generally to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive +1-approximation of all vertex eccentricities in deterministic O(α² m) time. This is in sharp contrast with general m-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in O(m^{2-ε}) time for any ε > 0.
As important special cases of our main result, we derive an O(m^{3/2})-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an O(k³m^{3/2})-time algorithm for this problem on graphs of asteroidal number at most k. Both results extend prior works on exact and approximate diameter computation within AT-free graphs. To the best of our knowledge, this is also the first deterministic subquadratic-time algorithm for computing the diameter within the subclasses of: chordal graphs of bounded leafage (generalizing the interval graphs), k-moplex graphs and k-polygon graphs (generalizing the permutation graphs) for any fixed k. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.
Our approach is purely combinatorial, that differs from most prior recent works in this area which have relied on geometric primitives such as Voronoi diagrams or range queries. On our way, we uncover interesting connections between the diameter problem, Lexicographic Breadth-First Search, graph extremities and the asteroidal number.

Guillaume Ducoffe. Obstructions to Faster Diameter Computation: Asteroidal Sets. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ducoffe:LIPIcs.IPEC.2022.10, author = {Ducoffe, Guillaume}, title = {{Obstructions to Faster Diameter Computation: Asteroidal Sets}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {10:1--10:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.10}, URN = {urn:nbn:de:0030-drops-173668}, doi = {10.4230/LIPIcs.IPEC.2022.10}, annote = {Keywords: Diameter computation, Asteroidal number, LexBFS} }

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

On sparse graphs, Roditty and Williams [2013] proved that no O(n^{2-ε})-time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier?
We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n^{1.6456}log^{O(1)} n).

Pierre Bergé, Guillaume Ducoffe, and Michel Habib. Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berge_et_al:LIPIcs.STACS.2022.9, author = {Berg\'{e}, Pierre and Ducoffe, Guillaume and Habib, Michel}, title = {{Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {9:1--9:21}, 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.2022.9}, URN = {urn:nbn:de:0030-drops-158192}, doi = {10.4230/LIPIcs.STACS.2022.9}, annote = {Keywords: Diameter, Eccentricities, Metric graph theory, Median graphs, Hypercubes} }

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

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in Õ(f(k) ⋅ (n+m)) time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and P₄-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle’s theorem to this problem directly, because a matching cannot be expressed in MSO₁ logic.
Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. It can be used to also compute the Edmonds-Gallai decomposition. For that, we do apply Courcelle’s theorem, but in order to compute the cardinality of a maximum matching rather than the matching itself, via the classic Tutte-Berge formula. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs based on the existence of balanced edge-cuts with bounded neighbourhood diversity, and with a distributed version of Courcelle’s theorem (Courcelle and Vanicat, DAM 2016) - of which we present here a slightly stronger version than the standard one in the literature - in order to evaluate the Tutte-Berge formula on various subgraphs of the input.
Finally, for the bipartite graphs of clique-width at most k, we present an alternative Õ(k²⋅(n+m))-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC'21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.

Guillaume Ducoffe. Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ducoffe:LIPIcs.IPEC.2021.15, author = {Ducoffe, Guillaume}, title = {{Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {15:1--15:17}, 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.2021.15}, URN = {urn:nbn:de:0030-drops-153987}, doi = {10.4230/LIPIcs.IPEC.2021.15}, annote = {Keywords: Maximum Matching, Maximum b-matching, Clique-width, Tree-width, Courcelle’s theorem, FPT in P} }

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

Given an n-vertex m-edge graph G of clique-width at most k, and a corresponding k-expression, we present algorithms for computing some well-known centrality indices (eccentricity and closeness) that run in O(2^{O(k)}(n+m)^{1+ε}) time for any ε > 0. Doing so, we can solve various distance problems within the same amount of time, including: the diameter, the center, the Wiener index and the median set. Our run-times match conditional lower bounds of Coudert et al. (SODA'18) under the Strong Exponential-Time Hypothesis. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using O(klog²{n}) bits per vertex and constructible in Õ(k(n+m)) time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an Õ(kn²)-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k, being given a k-expression. This partially answers an open question of Kratsch and Nelles (STACS'20). Our algorithms work for graphs with non-negative vertex-weights, under two different types of distances studied in the literature. For that, we introduce a new type of orthogonal range query as a side contribution of this work, that might be of independent interest.

Guillaume Ducoffe. Optimal Centrality Computations Within Bounded Clique-Width Graphs. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ducoffe:LIPIcs.IPEC.2021.16, author = {Ducoffe, Guillaume}, title = {{Optimal Centrality Computations Within Bounded Clique-Width Graphs}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {16:1--16:16}, 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.2021.16}, URN = {urn:nbn:de:0030-drops-153995}, doi = {10.4230/LIPIcs.IPEC.2021.16}, annote = {Keywords: Clique-width, Centralities computation, Facility Location problems, Distance-labeling scheme, Fine-grained complexity in P, Graph theory} }

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

We present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows:
- We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5-ε}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs.
- Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an Õ(n^{5/3})-time algorithm for computing all the eccentricities).
- We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2.
- Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, Bergé and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem. Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate.

Guillaume Ducoffe. Isometric Embeddings in Trees and Their Use in Distance Problems. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ducoffe:LIPIcs.MFCS.2021.43, author = {Ducoffe, Guillaume}, title = {{Isometric Embeddings in Trees and Their Use in Distance Problems}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {43:1--43:16}, 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.2021.43}, URN = {urn:nbn:de:0030-drops-144835}, doi = {10.4230/LIPIcs.MFCS.2021.43}, annote = {Keywords: Tree embeddings, Range queries, Centroid decomposition, Heavy-path decomposition, Diameter, Radius and all Eccentricities computations} }

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

The Wiener index of a graph G is the sum of all its distances. Up to renormalization, it is also the average distance in G. The problem of computing this parameter has different applications in chemistry and networks. We here study when it can be done in truly subquadratic time (in the size n+m of the input) on n-vertex m-edge graphs. Our main result is a complete answer to this question, assuming the Strong Exponential-Time Hypothesis (SETH), for all the hereditary subclasses of chordal graphs. Interestingly, the exact same result also holds for the diameter problem. The case of non-hereditary chordal subclasses happens to be more challenging. For the chordal Helly graphs we propose an intricate Õ(m^{3/2})-time algorithm for computing the Wiener index, where m denotes the number of edges. We complete our results with the first known linear-time algorithm for this problem on the dually chordal graphs. The former algorithm also computes the median set.

Guillaume Ducoffe. On Computing the Average Distance for Some Chordal-Like Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 44:1-44:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ducoffe:LIPIcs.MFCS.2021.44, author = {Ducoffe, Guillaume}, title = {{On Computing the Average Distance for Some Chordal-Like Graphs}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {44:1--44:16}, 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.2021.44}, URN = {urn:nbn:de:0030-drops-144841}, doi = {10.4230/LIPIcs.MFCS.2021.44}, annote = {Keywords: Wiener index, Graph diameter, Hardness in P, Chordal graphs, Helly graphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Given an n-vertex m-edge graph G with non-negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs:
- For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, in O~(n^{5/3}+m)-time, a cycle of weight at most twice the girth of G. This matches both the approximation factor and - almost - the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs.
- Then, we turn our algorithm into a deterministic (2+epsilon)-approximation for graphs with arbitrary non-negative edge-weights, at the price of a slightly worse running-time in O~(n^{5/3}polylog(1/epsilon)+m). For that, we introduce a generic method in order to obtain a polynomial-factor approximation of the girth in subquadratic time, that may be of independent interest.
- Finally, if we assume that the adjacency lists are sorted then we can get rid off the dependency in the number m of edges. Namely, we can transform our algorithms into an O~(n^{5/3})-time randomized 4-approximation for graphs with non-negative edge-weights. This can be derandomized, thereby leading to an O~(n^{5/3})-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and an O~(n^{5/3}polylog(1/epsilon))-time deterministic (4+epsilon)-approximation for graphs with non-negative edge-weights.
To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.

Guillaume Ducoffe. Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ducoffe:LIPIcs.ICALP.2019.49, author = {Ducoffe, Guillaume}, title = {{Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {49:1--49:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.49}, URN = {urn:nbn:de:0030-drops-106254}, doi = {10.4230/LIPIcs.ICALP.2019.49}, annote = {Keywords: girth, weighted graphs, approximation algorithms} }

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**Published in:** OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

A well-known problem for which it is difficult to improve the textbook algorithm is computing the graph diameter. We present two versions of a simple algorithm (one being Monte Carlo and the other deterministic) that for every fixed h and unweighted undirected graph G with n vertices and m edges, either correctly concludes that diam(G) < hn or outputs diam(G), in time O(m+n^{1+o(1)}). The algorithm combines a simple randomized strategy for this problem (Damaschke, IWOCA'16) with a popular framework for computing graph distances that is based on range trees (Cabello and Knauer, Computational Geometry'09). We also prove that under the Strong Exponential Time Hypothesis (SETH), we cannot compute the diameter of a given n-vertex graph in truly subquadratic time, even if the diameter is an Theta(n/log{n}).

Guillaume Ducoffe. A New Application of Orthogonal Range Searching for Computing Giant Graph Diameters. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 12:1-12:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ducoffe:OASIcs.SOSA.2019.12, author = {Ducoffe, Guillaume}, title = {{A New Application of Orthogonal Range Searching for Computing Giant Graph Diameters}}, booktitle = {2nd Symposium on Simplicity in Algorithms (SOSA 2019)}, pages = {12:1--12:7}, series = {Open Access Series in Informatics (OASIcs)}, ISBN = {978-3-95977-099-6}, ISSN = {2190-6807}, year = {2019}, volume = {69}, editor = {Fineman, Jeremy T. and Mitzenmacher, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.12}, URN = {urn:nbn:de:0030-drops-100383}, doi = {10.4230/OASIcs.SOSA.2019.12}, annote = {Keywords: Graph diameter, Orthogonal Range Queries, Hardness in P, FPT in P} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA'18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes.

Guillaume Ducoffe and Alexandru Popa. The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ducoffe_et_al:LIPIcs.ISAAC.2018.6, author = {Ducoffe, Guillaume and Popa, Alexandru}, title = {{The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.6}, URN = {urn:nbn:de:0030-drops-99549}, doi = {10.4230/LIPIcs.ISAAC.2018.6}, annote = {Keywords: maximum matching, FPT in P, modular decomposition, pruned graphs, one-vertex extensions, P\underline4-structure} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log^2{k})*(m+n) * log{n})-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.

Guillaume Ducoffe and Alexandru Popa. The b-Matching Problem in Distance-Hereditary Graphs and Beyond. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ducoffe_et_al:LIPIcs.ISAAC.2018.30, author = {Ducoffe, Guillaume and Popa, Alexandru}, title = {{The b-Matching Problem in Distance-Hereditary Graphs and Beyond}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {30:1--30:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.30}, URN = {urn:nbn:de:0030-drops-99783}, doi = {10.4230/LIPIcs.ISAAC.2018.30}, annote = {Keywords: maximum-cardinality matching, b-matching, FPT in P, split decomposition, distance-hereditary graphs} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. Our main contribution in this note is a new characterization of hyperbolicity for graphs (and for complete geodesic metric spaces). This characterization has algorithmic implications in the field of large-scale network analysis, which was one of our initial motivations. A sharp estimate of graph hyperbolicity is useful, {e.g.}, in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomial-time, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm for computing the hyperbolicity of an n-vertex graph G=(V,E) in optimal time O(n^2) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space.

Jérémie Chalopin, Victor Chepoi, Feodor F. Dragan, Guillaume Ducoffe, Abdulhakeem Mohammed, and Yann Vaxès. Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chalopin_et_al:LIPIcs.SoCG.2018.22, author = {Chalopin, J\'{e}r\'{e}mie and Chepoi, Victor and Dragan, Feodor F. and Ducoffe, Guillaume and Mohammed, Abdulhakeem and Vax\`{e}s, Yann}, title = {{Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.22}, URN = {urn:nbn:de:0030-drops-87356}, doi = {10.4230/LIPIcs.SoCG.2018.22}, annote = {Keywords: Gromov hyperbolicity, Graphs, Geodesic metric spaces, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

We consider a community formation problem in social networks, where the users are either friends or enemies. The users are partitioned into conflict-free groups (i.e., independent sets in the conflict graph G^- =(V,E) that represents the enmities between users). The dynamics goes on as long as there exists any set of at most k users, k being any fixed parameter, that can change their current groups in the partition simultaneously, in such a way that they all strictly increase their utilities (number of friends i.e., the cardinality of their respective groups minus one). Previously, the best-known upper-bounds on the maximum time of convergence were O(|V|alpha(G^-)) for k <= 2 and O(|V|^3) for k=3, with alpha(G^-) being the independence number of G^-. Our first contribution in this paper consists in reinterpreting the initial problem as the study of a dominance ordering over the vectors of integer partitions. With this approach, we obtain for k <= 2 the tight upper-bound O(|V| min {alpha(G^-), sqrt{|V|}}) and, when G^- is the empty graph, the exact value of order ((2|V|)^{3/2})/3. The time of convergence, for any fixed k >= 4, was conjectured to be polynomial [Escoffier et al., 2012][Kleinberg and Ligett, 2013]. In this paper we disprove this. Specifically, we prove that for any k >= 4, the maximum time of convergence is an Omega(|V|^{Theta(log{|V|})}).

Jean-Claude Bermond, Augustin Chaintreau, Guillaume Ducoffe, and Dorian Mazauric. How long does it take for all users in a social network to choose their communities?. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bermond_et_al:LIPIcs.FUN.2018.6, author = {Bermond, Jean-Claude and Chaintreau, Augustin and Ducoffe, Guillaume and Mazauric, Dorian}, title = {{How long does it take for all users in a social network to choose their communities?}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {6:1--6:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.6}, URN = {urn:nbn:de:0030-drops-87972}, doi = {10.4230/LIPIcs.FUN.2018.6}, annote = {Keywords: communities, social networks, integer partitions, coloring games, graphs} }

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