Efficient spectral graph sparsification via Krylov-subspace based spectral perturbation analysis

Abstract

Spectral graph sparsification aims to find an ultra-sparsified matrix which can be used as a good preconditioner for the original matrix. Low-stretch spanning tree could be constructed for this purpose. The relative condition number with the low-stretch spanning tree is bounded, which ensures the convergence of the PCG method with low-stretch spanning tree as preconditioner. Recently, spectral perturbation analysis is proposed to add a group of spectral critical edges to the spanning tree. The relative condition number can thus be further reduced. In this paper, we proposed a Krylov-subspace based spectral perturbation analysis to find the spectral critical edges. Compared with the traditional power-iteration-based approach, our proposed method can significantly improve the convergence of spectral perturbation analysis and thus accelerate the procedure of constructing the ultra-sparsified preconditioner.