Anytime and Distributed Approaches for Graph Matching

Due to the inherent genericity of graph-based representations, and thanks to the improvement of computer capacities, structural representations have become more and more popular in the field of Pattern Recognition (PR). In a graph-based representation, vertices and their attributes describe objects (or part of them) while edges represent interrelationships between the objects. Representing objects by graphs turns the problem of object comparison into graph matching (GM) where correspondences between vertices and edges of two graphs have to be found.In the domain of GM, over the last decade, Graph Edit Distance (GED) has been given a specific attention due to its flexibility to match many types of graphs. GED has been applied to a wide range of specific applications from molecule recognition to image classification. Researchers have shed light on the approximate methods that can find suboptimal solutions hopefully close to the optimal ones but the gap between optimal and suboptimal solutions has not been deeply studied yet. For that reason, in this thesis, we focus on exact GED algorithms. Unfortunately, exact GED methods have an exponential complexity. Thus, coming up with an exact GED algorithm that can be scaled up to match graphs involved in PR tasks is a great challenge. Two promising ways to cut-off computational time are search space pruning and distributed algorithms. To this end, we first propose a depth-first GED algorithm which requires less memory and search time. An evaluation of all possible solutions is performed without explicitly enumerating all of them. Candidates are discarded using an upper and lower bounds strategy.To find a trade-off between speed and optimality, we describe how to convert the proposed depth-first GED method into an anytime one that is capable of delivering a first solution very quickly. It also can find a list of improved solutions and eventually converges to the optimal solution instead of providing one and only one solution (i.e., the optimal solution). With the delight of more time, anytime methods can also reach the optimal solution. To illustrate the usage of anytime GM algorithms, we convert our depth-first GED algorithm into an anytime one. We analyze the properties of such methods to solve GM problems and consider the performance in terms of accuracy of the provided solution compared to the optimal or the best one found by a state-of-the-art methods.This thesis is also considered as a first attempt to reduce the run time of exact GED methods usingparallel and distributed fashions. Two parallel and distributed GED approaches are put forward; both of them are based on the depth-first GED method. The search space is decomposed into smaller search trees which are solved independently in a parallel or a distributed manner.To benchmark the proposed GED methods, we propose not only assessing GED methods in a classification context but also evaluating them in a graph-level one (i.e., evaluating their distance and matchin accuracy). Due to the exponential complexity of exact GED algorithms and in order to obtain this kind of information about methods, we propose analyzing the behavior of the eight compared methods under time and memory constraints. In addition to the performance evaluations metrics, we propose a graph database repository dedicated to GED. In this repository, we add graph-level information to well-known and publicly used databases. Added information consists of the best found edit distance of each pair of graphs as well as their vertex-to-vertex and edge-to-edge mappings corresponding to the best found distance. This information helps in assessing the feasibility of exact and approximate GED methods. This thesis brings into question the usual evidences saying that it is impossible to use exact errortolerant GM methods in real-world applications when matching large graphs, or even in a classification context. However, we argue and show that a new type of GM, referred to as anytime methods, can be successful in a graph-level context as well as a classification one. Anytime videos, pseudo-codes and the publications related to the thesis are publicly available at: http://www.rfai.li.univ-tours.fr/ PagesPerso/zabuaisheh/home.html. The thesis is also publicly available at: http://www.rfai.li.univ-tours.fr/Documents/Articles_RFAI/PhD2016zeina.pd

Un algorithme Hongrois pour l'appariement de graphes avec correction d'erreurs

International audienceBipartite 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))

Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back

We present an $\tilde{O}(m^{10/7})=\tilde{O}(m^{1.43})$-time algorithm for the maximum s-t flow and the minimum s-t cut problems in directed graphs with unit capacities. This is the first improvement over the sparse-graph case of the long-standing $O(m \min(\sqrt{m},n^{2/3}))$ time bound due to Even and Tarjan [EvenT75]. By well-known reductions, this also establishes an $\tilde{O}(m^{10/7})$-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated celebrated $O(m \sqrt{n})$ time bound of Hopcroft and Karp [HK73] whenever the input graph is sufficiently sparse