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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+��}) time for any �� > 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(klog��{n}) bits per vertex and constructible in O��(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 O��(kn��)-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., LIPIcs, Vol. 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), pages 16:1-16:16

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in O��(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 P���-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 MSO��� 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 O��(k�����(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., LIPIcs, Vol. 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021), pages 15:1-15:17

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width., Comment: Conference version appeared in ESA 2020. This is the full version which has been accepted for publication at SIDMA

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determine one new class of $(H,\overline{H})$-free graphs of bounded clique-width (as a side effect, this leaves only six classes of $(H_1,H_2)$-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the $|{\cal H}|=2$ case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set ${\cal F}$ of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for $(\{H,\overline{H}\}\cup {\cal F})$-free graphs coincides with the one for the $|{\cal H}|=2$ case if and only if ${\cal F}$ does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are $P_1$ and $P_4$, and $P_4$-free graphs have clique-width at most $2$). Finally, we discuss the consequences of our results for the Colouring problem., Comment: 39 pages, 7 figures

In this paper we study the problem of finding a small safe set S in a graph G, i.e. a non-empty set of vertices such that no connected component of G[S] is adjacent to a larger component in \(G - S\). We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth \(\mathsf {pw}\) and cannot be solved in time \(n^{o(\mathsf {pw})}\) unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number \(\mathsf {vc}\) unless \(\mathrm {PH} = \varSigma ^{\mathrm {p}}_{3}\), but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity \(\mathsf {nd}\), and (4) it can be solved in time \(n^{f(\mathsf {cw})}\) for some double exponential function f where \(\mathsf {cw}\) is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.

In the last decades, considerable efforts have been spent to characterize what makes NP-hard problems tractable. A successful approach in this line of research is the theory of parameterized complexity introduced by Downey and Fellows in the nineties.In this framework, the complexity of a problem is not measured only in terms of the input size, but also in terms of a parameter on the input. One of the most well-studied parameters is tree-width, a graph parameter which measures how close a graph is to the topological structure of a tree.It appears that tree-width has numerous structural properties and algorithmic applications.However, only sparse graph classes can have bounded tree-width. But, many NP-hard problems are tractable on dense graph classes. Most of the time, this tractability can be explained by the ability of these graphs to be recursively decomposable along vertex bipartitions (A,B) where the adjacency between A and B is simple to describe. A lot of graph parameters -- called width measures -- have been defined to characterize this ability, the most remarkable ones are certainly clique-width, rank-width, and mim-width. In this thesis, we study the algorithmic properties of these width measures. We provide a framework that generalizes and simplifies the tools developed for tree-width and for problems with a constraint of acyclicity or connectivity such as Connected Vertex Cover, Connected Dominating Set, Feedback Vertex Set, etc. For all these problems, we obtain 2^{O(k)}⋅n^{O(1)}, 2^{O(k log(k))}⋅n^{O(1)}, 2^{O(k²)}⋅n^{O(1)} and n^{O(k)} time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and mim-width. We also prove that there exists an algorithm solving Hamiltonian Cycle in time n^{O(k)}, when a clique-width decomposition of width k is given. Finally, we prove that we can count in polynomial time the minimal transversals of β-acyclic hypergraphs and the minimal dominating sets of strongly chordal graphs. All these results offer promising perspectives towards a generalization of width measures and their algorithmic applications., Durant ces dernières décennies, d'importants efforts et beaucoup de café ont été dépensés en vue de caractériser les instances faciles des problèmes NP-difficiles. Dans ce domaine de recherche, une approche s'avère être redoutablement efficace : la théorie de la complexité paramétrée introduite par Downey et Fellows dans les années 90.Dans cette théorie, la complexité d'un problème n'est plus mesurée uniquement en fonction de la taille de l'instance, mais aussi en fonction d'un paramètre. Dans cette boite à outils, la largeur arborescente est sans nul doute un des paramètres de graphe les plus étudiés. Ce paramètre mesure à quel point un graphe est proche de la structure topologique d'un arbre.La largeur arborescente a de nombreuses propriétés algorithmiques et structurelles.Néanmoins, malgré l'immense intérêt suscité par la largeur arborescente, seules les classes de graphes peu denses peuvent avoir une largeur arborescente bornée. Mais, de nombreux problèmes NP-difficiles s'avèrent faciles dans des classes de graphes denses. La plupart du temps, cela peut s'expliquer par l'aptitude de ces graphes à se décomposer récursivement en bipartitions de sommets (A,B) où le voisinage entre A et B possède une structure simple. De nombreux paramètres -- appelés largeurs -- ont été introduits pour caractériser cette aptitude, les plus remarquables sont certainement la largeur de clique , la largeur de rang, la largeur booléenne et la largeur de couplage induit. Dans cette thèse, nous étudions les propriétés algorithmiques de ces largeurs. Nous proposons une méthode qui généralise et simplifie les outils développés pour la largeur arborescente et les problèmes admettant une contrainte d'acyclicité ou de connexité tel que Couverture Connexe, Dominant Connexe, Coupe Cycle, etc. Pour tous ces problèmes, nous obtenons des algorithmes s'exécutant en temps 2^{O(k)}⋅n^{O(1)}, 2^{O(k log(k))}⋅n^{O(1)}, 2^{O(k²)}⋅n^{O(1)} et n^{O(k)} avec k étant, respectivement, la largeur de clique, la largeur de Q-rang, la larguer de rang et la largueur de couplage induit. On prouve aussi qu'il existe un algorithme pour Cycle Hamiltonien s'exécutant en temps n^{O(k)} quand une décomposition de largeur de clique k est donné en entrée. Finalement, nous prouvons qu'on peut compter en temps polynomial le nombre de transversaux minimaux d'hypergraphes β-acyclique ainsi que le nombre de dominants minimaux de graphes fortement triangulés. Tous ces résultats offrent des pistes prometteuses en vue d'une généralisation des largeurs et de leurs applications algorithmiques.

Graph decompositions are important for algorithmic purposes and for graph structure theory. We relate the split decomposition introduced by Cunnigham to vertex substitution, graph grammars and clique-width. For this purpose, we extend the usual notion of substitution, upon which modular decomposition is based, by considering graphs with dead (or non-boundary) vertices. We obtain a graph grammar for distance-hereditary graphs consisting of four rules collected in a single equation. We also bound the clique-width of a graph in terms of those of the components of a split decomposition that need not be canonical. For extending these results to directed graphs and their split decompositions (that we handle formally as graph-labelled trees), we need another extension of substitution : instead of two types of vertices, dead or alive as for undirected graphs, we need four types, in order to encode edge directions. We bound linearly the clique-width of a directed graph G in terms of the maximal clique-width of a component arising in a graph-labelled tree that defines G . This result concerns all directed graphs, not only the strongly connected ones considered by Cunningham.

This paper introduces the Bag of Graphs (BoG), a Bag-of-Words model that encodes in graphs the local structures of a digital object. We present a formal definition, introducing concepts and rules that make this model flexible and adaptable for different applications. We define two BoG-based methods – Bag of Singleton Graphs (BoSG) and Bag of Visual Graphs (BoVG), which create vector representations for graphs and images, respectively. We evaluate the Bag of Singleton Graphs (BoSG) for graph classification on four datasets of the IAM repository, obtaining significant results in accuracy and execution time. The method Bag of Visual Graphs (BoVG) is evaluated for image classification on Caltech and ALOI datasets, and for remote sensing image classification on images of Monte Santo and Campinas datasets. This framework opens possibilities for retrieval, classification, and clustering tasks on large datasets that use graph-based representations impractical before due to the complexity of inexact graph matching.

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and $P\_4$-sparseness). We believe that our most important result is an ${\cal O}(k^4 \cdot n + m)$-time algorithm for computing a maximum matching where $k$ is either the modular-width or the $P\_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, $P\_4$-lite graphs, $P\_4$-extendible graphs and $P\_4$-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

Tree-width and clique-width are two important graph complexity measures that serve as parameters in many fixed-parameter tractable algorithms. We give two algorithms that transform tree-decompositions represented by normal trees into clique-width terms (a rooted tree is normal for a graph if its nodes are the vertices of the graph and every two adjacent vertices are on a path of the tree starting at the root). As a consequence, we obtain that, for certain classes of sparse graphs, clique-width is polynomially bounded in terms of tree-width. It is even linearly bounded for planar graphs and incidence graphs. These results are useful in the construction of model-checking algorithms for problems described by monadic second-order formulae, including those allowing edge set quantifications.

Fuzzy median graph is an important new concept that can represent a set of fuzzy graphs by a representative fuzzy graph prototype. However, the computation of a fuzzy median graph remains a computationally expensive task. In this paper, we propose a new approximate algorithm for the computation of the Fuzzy Generalized Median Graph (FGMG) based on Fuzzy Attributed Relational Graph (FARG) embedding in a suitable vector space in order to capture the maximum information in graphs and to improve the accuracy and speed of document image retrieval processing. In this study, we focus on the application of FGMGs to the Content-based Document Retrieval (CBDR) problem. Experiments on real and synthetic databases containing a large number of FARGs with large sizes show that a CBDR using the FGMG as a dataset representative yields better results than an exhaustive and sequential retrieval in terms of gains in accuracy and time processing.

International audience; Some recent algebraic approaches to graph transformation include a pullback construction involving the match, that allows one to specify the cloning of items of the host graph. We pursue further this trend by proposing the Pullback-Pushout (pb-po) Approach, where we combine smoothly the classical modifications to a host graph specified by a rule (a span of graph morphisms) with the cloning of structures specified by another rule. The approach is shown to be a conservative extension of agree (and thus of the sqpo approach), and we show that it can be extended with standard techniques to attributed graphs. We discuss conditions to ensure a form of locality of transformations, and conditions to ensure that the attribution of transformed graphs is total.

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and $P_4$-sparseness). We believe that our most important result is an ${\cal O}(k^4 \cdot n + m)$-time algorithm for computing a maximum matching where $k$ is either the modular-width or the $P_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, $P_4$-lite graphs, $P_4$-extendible graphs and $P_4$-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

International audience; Bipartite graph matching algorithms become more and more popular to solve error-correcting graph matching problems and to approximate the graph edit distance of two graphs. However, the memory requirements and execution times of this method are respectively proportional to (n + m) 2 and (n + m) 3 where n and m are the order of the graphs. Subsequent developments reduced these complexities. However , these improvements are valid only under some constraints on the parameters of the graph edit distance. We propose in this paper a new formulation of the bipartite graph matching algorithm designed to solve efficiently the associated graph edit distance problem. The resulting algorithm requires O(nm) memory space and O(min(n, m) 2 max(n, m)) execution times.; L'appariement de graphes biparti deviennent de plus en plus populaires pour résoudre des problèmes d'appariement de graphes avec correction d'erreurs et pour approximer la distance d'édition sur graphes. Cependant, les exigences en mémoire et temps de calcul de cette méthode sont respectivement proportionnels à (n + m)^2 et (n + m)^3 où n et m représentent la taille des deux graphes. Des développements ultérieurs ont réduit ces complexités. Cependant, ces améliorations ne sont valables que sous certaines contraintes sur les paramètres de la distance d'édition. Nous proposons dans cet article une nouvelle formulation de l'algorithme Hongrois conçu pour résoudre efficacement le problème de distance d'édition associé. L'algorithme résultat nécessite un espace mémoire O (nm) et des temps d'exécution O (min (n, m)^2 max (n, m)).

Graph rewriting formalisms are well-established models for the representation of biological systems such as protein-protein interaction networks. The combinatorial complexity of these models usually prevents any explicit representation of the variables of the system, and one has to rely on stochastic simulations in order to sample the possible trajectories of the underlying Markov chain. The bottleneck of stochastic simulation algorithms is the update of the propensity function that describes the probability that a given rule is to be applied next. In this paper we present an algorithm based on a data structure, called extension basis, that can be used to update the counts of predefined graph observables after a rule of the model has been applied. Extension basis are obtained by static analysis of the graph rewriting rule set. It is derived from the construction of a qualitative domain for graphs and the correctness of the procedure is proven using a purely domain theoretic argument.

Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}\cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$ or worse.

Algorithms for finding dense regions in an input graph have proved to be effective tools in graph mining and data analysis. Recently, Tatti and Gionis [WWW 2015] presented a novel graph decomposition (known as the locally-dense decomposition) that is similar to the well-known k-core decomposition, with the additional property that its components are arranged in order of their densities. Such a decomposition provides a valuable tool in graph mining. Unfortunately, their algorithm for computing the exact decomposition is based on a maximum-flow algorithm which cannot scale to massive graphs, while the approximate decomposition defined by the same authors misses several interesting properties. This calls for scalable algorithms for computing such a decomposition. In our work, we devise an efficient algorithm which is able to compute exact locally-dense decompositions in real-world graphs containing up to billions of edges. Moreover, we provide a new definition of approximate locally-dense decomposition which retains most of the properties of an exact decomposition, for which we devise an algorithm that can scale to real-world graphs containing up to tens of billions of edges. Our algorithm is based on the classic Frank-Wolfe algorithm which is similar to gradient descent and can be efficiently implemented in most of the modern architectures dealing with massive graphs. We provide a rigorous study of our algorithms and their convergence rates. We conduct an extensive experimental evaluation on multi-core architectures showing that our algorithms converge much faster in practice than their worst-case analysis. Our algorithm is even more efficient for the more specialized problem of computing a densest subgraph.

International audience; Resolution is a proof-search method for proving unsatisfia-bility problems. Various refinements have been proposed to improve the efficiency of this method. However, when we try to prove some graph properties, it seems that none of the refinements have an efficiency comparable with traditional graph traversal algorithms. In this paper we propose a way of encoding some graph problems as resolution. We define a selection function and a new subsumption rule to avoid redundancies while solving such problems.

International audience; In this paper, we present a solution for executing bigraphical reactive systems based on an investigation on graph transformation systems. For this, we encode a bigraph into a ranked graph. This encoding is ensured, formally, by defining a faithful functor that allows to move from bigraph category to ranked graph category. Then, we show that reaction rules can be simulated with graph rules.

We present logically based methods for constructing XP and FPT graph algorithms, parametrized by tree-width or clique-width. We will use fly-automata introduced in a previous article. They make possible to check properties that are not monadic second-order expressible because their states may include counters, so that their sets of states may be infinite. We equip these automata with output functions, so that they can compute values associated with terms or graphs. Rather than new algorithmic results we present tools for constructing easily certain dynamic programming algorithms by combining predefined automata for basic functions and properties., Comment: Accepted for publication in Theoretical Computer Science

The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for $(\mbox{diamond},H)$-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for $(\mbox{diamond}, P_1+2P_2)$-free graphs. Our technique for handling this case is to reduce the graph under consideration to a $k$-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of $(H_1,H_2)$-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of $(H_1,H_2)$-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8., Comment: 30 pages, 3 figures. An extended abstract of this paper was published in the proceedings of SWAT 2016 (DOI:10.4230/LIPIcs.SWAT.2016.16)

© 2015 Elsevier B.V.. Graphs are flexible and powerful representations for non-vectorial structured data. Graph kernels have been shown to enable efficient and accurate statistical learning on this important domain, but many graph kernel algorithms have high order polynomial time complexity. Efficient graph kernels rely on a discrete node labeling as a central assumption. However, many real world domains are naturally described by continuous or vector valued node labels. In this paper, we propose an efficient graph representation and comparison scheme for large graphs with continuous vector labels, the pyramid quantized Weisfeiler-Lehman graph representation. Our algorithm considers statistics of subtree patterns with discrete labels based on the Weisfeiler-Lehman algorithm and uses a pyramid quantization strategy to determine a logarithmic number of discrete labelings that results in a representation that guarantees a multiplicative error bound on an approximation to the optimal partial matching. As a result, we approximate a graph representation with continuous vector labels as a sequence of graphs with increasingly granular discrete labels. We evaluate our proposed algorithm on two different tasks with real datasets, on a fMRI analysis task and on the generic problem of 3D shape classification. Source code of the implementation can be downloaded from https://web.imis.athena-innovation.gr/~kgkirtzou/Projects/WLpyramid.html. publisher: Elsevier articletitle: The pyramid quantized Weisfeiler–Lehman graph representation journaltitle: Neurocomputing articlelink: http://dx.doi.org/10.1016/j.neucom.2015.09.023 content_type: article copyright: Copyright © 2015 Elsevier B.V. All rights reserved. ispartof: Neurocomputing vol:173 issue:3 pages:1495-1507 status: published

A more descriptive but too long title would be : Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute "on the fly" the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quan-tifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective., Submitted for publication in December 2015

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a $(1-\epsilon)$-approximation in time $f(k,\epsilon)\cdot poly(n)$. 3. There is no FPTAS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results., Comment: Appeared in IPEC 2015

We survey the use and effect of decomposition-based techniques in qualitative constraint-based reasoning, and clarify the notions of a tree decomposition, a chordal graph, and a partitioning graph, and their implication with a particular constraint property that has been extensively used in literature, namely, patchwork. As a consequence, we prove that a recently proposed decomposition-based approach that was presented in [AAAI, 2014 ] for checking the satisfiability of qualitative spatial constraint networks lacks soundness. Therefore, the approach becomes quite controversial as it does not seem to offer any technical advance at all, while experimental evaluation of it in a following paper presented in [ICTAI, 2014 ] becomes questionable.

International audience; The verification of monadic second-order (MSO) graph properties, equivalently, the model-checking problem for MSO logic over finite binary relational structures , is fixed-parameter tractable (FPT) for the parameter consisting of the formula that expresses the property and the tree-width or the clique-width of the input graph or structure. How to build usable algorithms for this problem ? We use for that fly-automata : the input is a term that describes the graph and the automaton computes, for each term, the necessary transitions and only these ones.

We show that for every forest T the linear rank-width of T is equal to the path-width of T, and we show that the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k.

International audience; Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, preserve the tree structures and that they are computable in polynomial time. We apply our characterisation to linear clique-width. And we relate our characterisation to a clique-width characterisation by Heule and Szeider that is used to reduce the clique-width decision problem to a satisfiability problem.

The Recognizability Theorem states that if a set of finite graphs is definable by a monadic second-order (MSO) sentence, then it is recognizable with respect to the graph algebra upon which the definition of clique-width is based. Recognizability is an algebraic notion, defined in terms of congruences that can also be formulated by means of finite automata on the terms that describe the considered graphs. This theorem entails that the verification of MSO graph properties, or equivalently, the model-checking problem for MSO logic over finite binary relational structures, is fixed-parameter tractable (FPT) for the parameter consisting of the formula that expresses the property and the clique-width (or the tree-width) of the input graph or structure. The corresponding algorithms can be implemented by means of fly-automata whose transitions are computed on the fly and not tabulated. We review two versions of recognizability, we present fly-automata by means of examples showing that they can also compute values attached to graphs. We show that fly-automata with infinite sets of states yield a simple proof of the strong version of the Recognizability Theorem. This proof has not been published previously.

International audience; Chemoinformatics aims to predict molecular properties using informational methods. Computer science's research fields concerned by this domain are machine learning and graph theory. An interesting approach consists in using graph kernels which allow to combine graph theory and machine learning frameworks. Graph kernels allow to define a similarity measure between molecular graphs corresponding to a scalar product in some Hilbert space. Most of existing graph kernels proposed in chemoinformatics do not allow to explicitly encode cyclic information, hence limiting the efficiency of these approaches. In this paper, we propose to define a cyclic representation encoding the relative positioning of substituents around a cycle. We also propose a graph kernel taking into account this information. This contribution has been tested on three classification problems proposed in chemoinformatics.

International audience; The main objective of this paper is to add one more brick in building the CG model as a knowledge representation model autonomous from logic. The CG model is not only a graphical representation of logic, it is much more: it is a declarative model encoding knowledge in a mathematical theory, namely labelled graph theory, which has efficient computable forms, with a fundamental graph operation on the encodings to do reasoning, projection, which is a labelled graph morphism. Main topics of this paper are: a generalized formalism for simple CGs; a strong equivalence between CSP (Constraint Satisfaction Problem) and labelled graph morphism. This correspondence allows the transportation of efficient algorithms from one domain to the other, and confirms that projection —or more generally labelled graph morphism— firmly moors CGs to combinatorial algorithmics, which is a cornerstone of computer science. The usual sound and complete first order logic semantics for CGs is still valid for our generalized model. This, plus the ease of doing important reasonings with CGs —for instance plausible reasonings by using some maximal join operations— without, at least for the moment, logical semantics, strengthens our belief that CGs must also be studied and developed independently from logic.

The Hadwiger number of a graph \(G\) is the largest integer \(h\) such that \(G\) has the complete graph \(K_h\) as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer \(h\) such that \(G\) has a minor with \(h\) vertices and diameter at most \(s\). We show that this problem can be solved in polynomial time on AT-free graphs when \(s\ge 2\), but is NP-hard on chordal graphs for every fixed \(s\ge 2\).

International audience; This paper studies the problem of learning from a set of input graphs, each of them representing a different relation over the same set of nodes. Our goal is to merge those input graphs by embedding them into an Euclidean space related to the commute time distance in the original graphs. This is done with the help of a small number of labeled nodes. Our algorithm output a combined kernel that can be used for different graph learning tasks. We consider two combination methods: the (classical) linear combination and the sigmoid combination. We compare the combination methods on node classification tasks using different semi-supervised graph learning algorithms. We note that the sigmoid combination method exhibits very positive results.

International audience; This paper develops methods to reason about graph transformation rules for proving the preservation of structural properties, especially global properties on reachability. We characterize a graph transformation rule with an applicability condition specifying the matching conditions of the rule on a host graph as well as the properties to be preserved during the transformation. Our previous work has demonstrated the possibility to reason about a graph transformation at rulelevel with applicability conditions restricted to Boolean combinations of edge expressions. We now extend the approach to handle the applicability conditions containing transitive closure of edges, which implicitly refer to an unbounded number of nodes. We show how these can be internalized into a finite pattern graph in order to enable verification of global properties on paths instead of local properties on edges only.

International audience; The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite automata from monadic second-order sentences. These automata recognize the terms over fixed finite signatures that define graphs satisfying the given sentences. However, this construction produces automata of hyper-exponential sizes, and is thus impossible to use in practice in many cases. To overcome this difficulty, we propose to specify the transitions of automata by programs instead of tables. Such automata are called fly-automata. By using them, we can check certain monadic second-order graph properties with limited quantifier alternation depth, that are nevertheless interesting for Graph Theory. We give explicit constructions of automata relative to graphs of bounded clique-width, and we report on experiments.

International audience; Graphs are the most powerful, expressive and convenient data structures but there is a lack of efficient computational tools and algorithms for processing them. The embedding of graphs into numeric vector spaces permits them to access the state-of-the-art computational efficient statistical models and tools. In this paper we take forward our work on explicit graph embedding and present an improvement to our earlier proposed method, named "fuzzy multilevel graph embedding - FMGE", through feature selection technique. FMGE achieves the embedding of attributed graphs into low dimensional vector spaces by performing a multilevel analysis of graphs and extracting a set of global, structural and elementary level features. Feature selection permits FMGE to select the subset of most discriminating features and to discard the confusing ones for underlying graph dataset. Experimental results for graph classification experimentation on IAM letter, GREC and fingerprint graph databases, show improvement in the performance of FMGE.

Community detection and evaluation is an important task in graph mining. In many cases, a community is defined as a subgraph characterized by dense connections or interactions between its nodes. A variety of measures are proposed to evaluate different quality aspects of such communities—in most cases ignoring the directed nature of edges. In this paper, we introduce novel metrics for evaluating the collaborative nature of directed graphs—a property not captured by the single node metrics or by other established community evaluation metrics. In order to accomplish this objective, we capitalize on the concept of graph degeneracy and define a novel D-core framework, extending the classic graph-theoretic notion of $$k$$ -cores for undirected graphs to directed ones. Based on the D-core, which essentially can be seen as a measure of the robustness of a community under degeneracy, we devise a wealth of novel metrics used to evaluate graph collaboration features of directed graphs. We applied the D-core approach on large synthetic and real-world graphs such as Wikipedia, DBLP, and ArXiv and report interesting results at the graph as well at the node level.

International audience; Graph rewriting systems have long been around in theoretical computer science, and are used as a computing model in discrete mathematics, algebra, mathematical logic or even linguistics. Graph rewriting systems operate on graphs by substituting local patterns. Starting from an initial graph, the system iterates transformation rules in a non necessarily deterministic way. The combination of rule applications turn the study of such a system into an involved task requiring high level expertise. The inherent complexity of graph rewriting systems motivated the use of graphical diagrams, even by experts, to convey ideas and theoretical concepts. In these situations, users are not experts in theoretical computer science but rather focus on how a graph rewriting system behaves as a model of the problem they study. Questions emerging from the target application nevertheless translate into questions about the rewriting system. The goal of this work is to allow users to experiment with a graph rewriting system, and reason about the system or the model it supports based on graphical representations of all its components. Visually reasoning about the system, domain experts are able to abstract and refine models, use simulation and deduction to compute or check postulated properties and make testable predictions about possible outcomes. PORGY possesses genuine features and appears to be the first system to offer multiple coordinated views, allowing users to visualize and manipulate the system at both a global and local scale.

International audience; In the framework of online object retrieval with learning, we address the problem of graph matching using kernel functions. An image is represented by a graph of regions where the edges represent the spatial relationships. Kernels on graphs are built from kernel on walks in the graph. This paper firstly proposes new kernels on graphs and on walks, which are very efficient for graphs of regions. Secondly we propose fast solutions for exact or approximate computation of these kernels. Thirdly we show results for the retrieval of images containing a specific object with the help of very few examples and counter-examples in the framework of an active retrieval scheme.

International audience; Graph cuts are widely used in many fields of computer vision in order to minimize in small polynomial time complexity certain classes of energies. These specific classes depend on the way chosen to build the graphs representing the problems to solve. We study here all possible ways of building graphs and the associated energies minimized, leading to the exhaustive family of energies minimizable exactly by a graph cut. To do this, we consider the issue of coding pixel labels as states of the graph, i.e. the choice of state interpretations. The family obtained comprises many new classes, in particular energies that do not satisfy the submodularity condition, including energies that are even not permuted-submodular. A generating subfamily is studied in details, in particular we propose a canonical form to represent Markov random fields, which proves useful to recognize energies in this subfamily in linear complexity almost surely, and then to build the associated graph in quasilinear time. A few experiments are performed, to illustrate the new possibilities offered.

International audience; One of the fundamental problems in graph databases is similarity search for graphs of interest. Existing approaches dealing with this problem rely on a single similarity measure between graph structures. In this paper, we suggest an alternative approach allowing for searching similar graphs to a graph query where similarity between graphs is rather modeled by a vector of scalars than a unique scalar. To this end, we introduce the notion of similarity skyline of a graph query defined by the subset of graphs of the target database that are the most similar to the query in a Pareto sense. The idea is to achieve a d-dimensional comparison between graphs in terms of local distance (or similarity) measures and to retrieve those graphs that are maximally similar in the sense of the Pareto dominance relation. A diversity-based method for refining the retrieval result is proposed as well.

International audience; Many graph visualization systems use graph hierarchies to organize a large input graph into logical components. These approaches detect features globally in the data and place these features inside levels of a hierarchy. However, this feature detection is a global process and does not consider nodes of the graph near a feature of interest. TugGraph is a system for exploring paths and proximity around nodes and subgraphs in a graph. The approach modifies a pre-existing hierarchy in order to see how a node or subgraph of interest extends out into the larger graph. It is guaranteed to create path-preserving hierarchies, so that the abstraction shown is meaningful with respect to the underlying structure of the graph. The system works well on graphs of hundreds of thousands of nodes and millions of edges. TugGraph is able to present views of this proximal information in the context of the entire graph in seconds, and does not require a layout of the full graph as input.

Given a graph G we consider the problem of preprocessing it so that given two vertices x,y and a set X of vertices, we can efficiently report the shortest path (or just its length) between x,y that avoids X. We attach labels to vertices in such a way that this length can be determined from the labels of x,y and the vertices X. For a graph with n vertices of tree-width or clique-width k, we construct labels of size O(k 2log 2 n). The constructions extend to directed graphs. The problem is motivated by routing in networks in case of failures or of routing policies which forbid certain paths.

International audience; We define NLC Fk to be the restriction of the class of graphs NLC k , where relabelling functions are exclusively taken from a set F of functions from {1,...,k} into {1,...,k}. We characterize the sets of functions F for which NLC Fk is well-quasi-ordered by the induced subgraph relation ≤ i . Precisely, these sets F are those which satisfy that for every f,g∈F , we have Im(f ∘ g) = Im(f) or Im(g ∘ f) = Im(g). To obtain this, we show that words (or trees) on F are well-quasi-ordered by a relation slightly more constrained than the usual subword (or subtree) relation. A class of graphs is n-well-quasi-ordered if the collection of its vertex-labellings into n colors forms a well-quasi-order under ≤ i , where ≤ i respects labels. Pouzet (C R Acad Sci, Paris Sér A-B 274:1677-1680, 1972) conjectured that a 2-well-quasi-ordered class closed under induced subgraph is in fact n-well-quasi-ordered for every n. A possible approach would be to characterize the 2-well-quasi-ordered classes of graphs. In this respect, we conjecture that such a class is always included in some well-quasi-ordered NLC Fk for some family F . This would imply Pouzet's conjecture.

International audience; This paper compares several metaheuristics on the balanced graph bisection problem and identifies their relative performance on structured test graphs. For this purpose, a Simple implementation of a graph Bisection Iterated Local Search algorithm (SBILS) is introduced. Its results are compared to the well-known Genetic Bisection Algorithm, to the state-of-the-art graph partitioning programs Metis and Scotch, and to the Fusion-Fission graph partitioning metaheuristic.

Graph constraints were introduced in the area of graph transformation, in connection with the notion of (negative) application conditions, as a form to limit the applicability of transformation rules. However, we believe that graph constraints may also play a significant role in the area of visual software modelling or in the specification and verification of semi-structured documents or websites (i.e. HTML or XML sets of documents). In this sense, after some discussion on these application areas, we concentrate on the problem of how to prove the consistency of specifications based on this kind of constraints. In particular, we present proof rules for two classes of graph constraints and show that our proof rules are sound and (refutationally) complete for each class. In addition, we study clause subsumption in this context as a form to speed up refutation.

International audience; Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very natural: If a parameter like the treewidth of a graph is small, algorithms based on dynamic programming perform well. On the other side, if the treewidth is large, then there must be vertices of high degree in the graph, which is good for branching algorithms. We give several examples of possible combinations of branching and programming which provide the fastest known algorithms for a number of NP hard problems. All our algorithms require non-trivial balancing of these two techniques. In the first approach the algorithm either performs fast branching, or if there is an obstacle for fast branching, this obstacle is used for the construction of a path decomposition of small width for the original graph. Using this approach we give the fastest known algorithms for Minimum Maximal Matching and for counting all 3-colorings of a graph. In the second approach the branching occurs until the algorithm reaches a subproblem with a small number of edges (and here the right choice of the size of subproblems is crucial) and then dynamic programming is applied on these subproblems of small width. We exemplify this approach by giving the fastest known algorithm to count all minimum weighted dominating sets of a graph. We also discuss how similar techniques can be used to design faster parameterized algorithms.

This tutorial will present an overview of the use of Monadic Second-Order Logic to describe sets of finite graphs and graph transformations, in relation with the notions of tree-width and clique-width. It will review applications to the construction of algorithms, to Graph Theory and to the extension to graphs of Formal Language Theory concepts.

In this paper, we study the relationship between the number of $n$-vertex graphs in a hereditary class $\cal X$, also known as the speed of the class $\cal X$, and boundedness of the clique-width in this class. We show that if the speed of $\cal X$ is faster than $n!c^n$ for any $c$, then the clique-width of graphs in $\cal X$ is unbounded, while if the speed does not exceed the Bell number $B_n$, then the clique-width is bounded by a constant. The situation in the range between these two extremes is more complicated. This area contains both classes of bounded and unbounded clique-width. Moreover, we show that classes of graphs of unbounded clique-width may have slower speed than classes where the clique-width is bounded.

International audience; In this article we present a new approach for the classification of structured data using graphs. We suggest to solve the problem of complexity in measuring the distance between graphs by using a new graph signature. We present an extension of the vector representation based on pattern frequency, which integrates labeling information. In this paper, we compare the results achieved on public graph databases for the classification of symbols and letters using this graph signature with those obtained using the graph edit distance.