Pursuing More Effective Graph Spectral Sparsifiers via Approximate Trace Reduction

Spectral graph sparsification aims to find ultra-sparse subgraphs which can preserve spectral properties of original graphs. In this paper, a new spectral criticality metric based on trace reduction is first introduced for identifying spectrally important off-subgraph edges. Then, a physics-inspired truncation strategy and an approach using approximate inverse of Cholesky factor are proposed to compute the approximate trace reduction efficiently. Combining them with the iterative densification scheme in \cite{feng2019grass} and the strategy of excluding spectrally similar off-subgraph edges in \cite{fegrass}, we develop a highly effective graph sparsification algorithm. The proposed method has been validated with various kinds of graphs. Experimental results show that it always produces sparsifiers with remarkably better quality than the state-of-the-art GRASS \cite{feng2019grass} in same computational cost, enabling more than 40% time reduction for preconditioned iterative equation solver on average. In the applications of power grid transient analysis and spectral graph partitioning, the derived iterative solver shows 3.3X or more advantages on runtime and memory cost, over the approach based on direct sparse solver.