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Laplacian matrices of graphs arise in large-scale computational applicationssuch as machine learning; spectral clustering of images, genetic data and webpages; transportation network flows; electrical resistor circuits; and ellipticpartial differential equations discretized on unstructured grids with finiteelements. A Lean Algebraic Multigrid (LAMG) solver of the linear system Ax=b ispresented, where A is a graph Laplacian. LAMG's run time and storage are linearin the number of graph edges. LAMG consists of a setup phase, in which asequence of increasingly-coarser Laplacian systems is constructed, and aniterative solve phase using multigrid cycles. General graphs pose algorithmicchallenges not encountered in traditional applications of algebraic multigrid.LAMG combines a lean piecewise-constant interpolation, judicious nodeaggregation based on a new node proximity definition, and an energy correctionof the coarse-level systems. This results in fast convergence and substantialoverhead and memory savings. A serial LAMG implementation scaled linearly for adiverse set of 1666 real-world graphs with up to six million edges. Thismultilevel methodology can be fully parallelized and extended to eigenvalueproblems and other graph computations.

Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver