Recursive Thresholding and Partial Coloring for Deterministic Spectral Sparsification

Li and Vaughn [LV26] gave a deterministic algorithm that constructs a $(1\pm\epsilon)$-spectral sparsifier with $O(n\epsilon^{-2}\log^{24+o(1)}n) $ edges in $O(m^{1+o(1)}+n^2\epsilon^{-9/2}\log^{113/2+o(1)}n) $  time for graphs whose edge-weight ratio is polynomially bounded.

We give three deterministic algorithms. Our first algorithm constructs a
$(1\pm\epsilon)$-spectral sparsifier with $O(n\epsilon^{-2}\log^{6+o(1)}n)$ edges in $O(m^{1+o(1)}+n^2\epsilon^{-4}\log^{20+o(1)}n)$ time. This improves their edge bound by a factor of $\log^{18}n$. 

Our second algorithm constructs a $(1\pm\epsilon)$-spectral sparsifier with $O(n\epsilon^{-2}\log n)$ edges in $O(m^{1+o(1)}+(n\epsilon^{-2})^{2+o(1)})$ time.

Our third algorithm constructs a $(1\pm\epsilon)$-spectral sparsifier with $O(n\epsilon^{-2})$ edges in time $  O(m^{1+o(1)}+(n\epsilon^{-2})^{\omega+1/2+o(1)}).$ Here $\omega$ denotes the exponent of matrix multiplication; the current best upper bound is $\omega<2.371177$ [DEK+26]. For comparison, the algorithm of Batson, Spielman, and Srivastava [BSS12] runs in $O(\epsilon^{-2}mn^3)$ time.