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Degree3 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 nodedisj ..."
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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 nodedisjoint paths, and computing minors seems easier in subcubic 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 polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph. 1
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 sousjacent. 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 methods 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, associated wi ..."
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Abstract. We consider the problem of designing efficient iterative methods 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, associated 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 preconditioners, known as tree preconditioners, introduced by Vaidya, that have been shown to lead to asymptotic speedup 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
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 optimize the intuitive notions of intracluster density and intercluster sparsity. In many realworld 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 optimize the intuitive notions of intracluster density and intercluster sparsity. In many realworld 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 structures 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 smallsize graph motifs, which are adopted as prime features to differentiate edges with varied cluster significance. Pathbased indexes and pathjoin algorithms are further designed to compute graphmotif based cluster significance of edges for graph sparsification. We perform experimental studies in realworld graphs, and results demonstrate that gSparsify can bring significant speedup to existing graph clustering methods with an improvement to graph clustering quality.
Degree3 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 nodedisj ..."
Abstract
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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 nodedisjoint paths, and computing minors seems easier in subcubic 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 polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph.