Spectral graph sparsification aims to find an ultra-sparse subgraph whose Laplacian matrix can well approximate the original Laplacian matrix in terms of its eigenvalues and eigenvectors. The resultant sparsified subgraph can be efficiently leveraged as a proxy in a variety of numerical computation applications and graph-based algorithms. This paper introduces a practically efficient, nearly-linear time spectral graph sparsification algorithm that can immediately lead to the development of nearly-linear time symmetric diagonally-dominant (SDD) matrix solvers. Our spectral graph sparsification algorithm can efficiently build an ultra-sparse subgraph from a spanning tree subgraph by adding a few "spectrally-critical" off-tree edges back to the spanning tree, which is enabled by a novel spectral perturbation approach and allows to approximately preserve key spectral properties of the original graph Laplacian. Extensive experimental results confirm the nearly-linear runtime scalability of an SDD matrix solver for large-scale, real-world problems, such as VLSI, thermal and finite-element analysis problems, etc. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within two minutes using a single CPU core and about 6 GB  memory.
['Zhuo Feng']
Spectral graph sparsification in nearly-linear time leveraging efficient spectral perturbation analysis