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Graph Sparsification Approaches for Laplacian Smoothing
"... Abstract Given a statistical estimation problem where regularization is performed according to the structure of a large, dense graph G, we consider fitting the statistical estimate using a sparsified surrogate graphG, which shares the vertices of G but has far fewer edges, and is thus more tractabl ..."
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Abstract Given a statistical estimation problem where regularization is performed according to the structure of a large, dense graph G, we consider fitting the statistical estimate using a sparsified surrogate graphG, which shares the vertices of G but has far fewer edges, and is thus more tractable to work with computationally. We examine three types of sparsification: spectral sparsification, which can be seen as the result of sampling edges from the graph with probabilities proportional to their effective resistances, and two simpler sparsifiers, which sample edges uniformly from the graph, either globally or locally. We provide strong theoretical and experimental results, demonstrating that sparsification before estimation can give statistically sensible solutions, with significant computational savings.
gSparsify: Graph Motif Based Sparsification for Graph Clustering
"... Graph clustering is a fundamental problem that partitions vertices of a graph into clusters with an objective to opti-mize the intuitive notions of intra-cluster density and inter-cluster sparsity. In many real-world applications, however, the sheer sizes and inherent complexity of graphs may render ..."
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Graph clustering is a fundamental problem that partitions vertices of a graph into clusters with an objective to opti-mize the intuitive notions of intra-cluster density and inter-cluster sparsity. In many real-world applications, however, the sheer sizes and inherent complexity of graphs may render existing graph clustering methods inefficient or incapable of yielding quality graph clusters. In this paper, we propose gSparsify, a graph sparsification method, to preferentially retain a small subset of edges from a graph which are more likely to be within clusters, while eliminating others with less or no structure correlation to clusters. The resultant simplified graph is succinct in size with core cluster struc-tures well preserved, thus enabling faster graph clustering without a compromise to clustering quality. We consider a quantitative approach to modeling the evidence that edges within densely knitted clusters are frequently involved in small-size graph motifs, which are adopted as prime features to differentiate edges with varied cluster significance. Path-based indexes and path-join algorithms are further designed to compute graph-motif based cluster significance of edges for graph sparsification. We perform experimental studies in real-world graphs, and results demonstrate that gSparsify can bring significant speedup to existing graph clustering methods with an improvement to graph clustering quality.
Algorithmique numérique Approximation creuse de graphes
"... L’état décide de changer sa stratégie d’investissement pour l’amélioration du réseau routier. Un projet de recherche a éte ́ lance ́ et l’équipe chargée d’optimiser la politique d’investissement a développe ́ un outil performant répondant a ̀ la question. Cependant, cet outil n’est pas cap ..."
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L’état décide de changer sa stratégie d’investissement pour l’amélioration du réseau routier. Un projet de recherche a éte ́ lance ́ et l’équipe chargée d’optimiser la politique d’investissement a développe ́ un outil performant répondant a ̀ la question. Cependant, cet outil n’est pas capable de travailler sur le réseau routier complet du ̂ a ̀ la complexite ́ du problème d’optimisation sous-jacent. Votre tache consiste a ̀ pallier ce problème en fournissant a ̀ l’équipe d’optimisation un réseau de taille réduite sur base du réseau complet. Dans ce graphe, un sommet représente un carrefour et une arête une route du réseau. A ̀ chaque arête est associe ́ un poids entre 0 et 1 représentant la densite ́ de la circulation sur la route. Un exemple de réseau routier a ̀ 30 routes est donne ́ a ̀ la Figure 1a. Son homologue creux est donne ́ a ̀ la Figure 1b. (a) Graphe initial (b) Graphe creux
Efficient and practical tree preconditioning for solving Laplacian systems
"... Abstract. We consider the problem of designing efficient iterative meth-ods for solving linear systems. In its full generality, this is one of the oldest problems in numerical analysis with a tremendous number of practical applications. We focus on a particular type of linear systems, associ-ated wi ..."
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Abstract. We consider the problem of designing efficient iterative meth-ods for solving linear systems. In its full generality, this is one of the oldest problems in numerical analysis with a tremendous number of practical applications. We focus on a particular type of linear systems, associ-ated with Laplacian matrices of undirected graphs, and study a class of iterative methods for which it is possible to speed up the convergence through combinatorial preconditioning. We consider a class of precondi-tioners, known as tree preconditioners, introduced by Vaidya, that have been shown to lead to asymptotic speed-up in certain cases. Rather than trying to improve the structure of the trees used in preconditioning, we propose a very simple modification to the basic tree preconditioner, which can significantly improve the performance of the iterative linear solvers in practice. We show that our modification leads to better conditioning for some special graphs, and provide extensive experimental evidence for the decrease in the complexity of the preconditioned conjugate gradient method for several graphs, including 3D meshes and complex networks. 1
Degree-3 Treewidth Sparsifiers
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, |V (H) | is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node-disj ..."
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, |V (H) | is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) |V (H) | = O(k4); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in |V (G) | and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.
