A cascadic multigrid algorithm for computing the Fiedler vector of graph Laplacians

John C. Urschel, Jinchao Xu, Xiaozhe Hu, Ludmil T. Zikatanov

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Abstract

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.

Original languageEnglish (US)
Pages (from-to)209-226
Number of pages18
JournalJournal of Computational Mathematics
Volume33
Issue number2
DOIs
StatePublished - Mar 1 2015

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All Science Journal Classification (ASJC) codes

  • Computational Mathematics

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