Determinant-Preserving Sparsification of SDDM Matrices

We show that variants of spectral sparsification routines can preserve the total spanning tree counts of graphs. By Kirchhoff's matrix-tree theorem, this is equivalent to preserving the determinant of a graph Laplacian minor or, equivalently, of any symmetric diagonally dominant matrix (SDDM). Our analyses utilize this combinatorial connection to bridge the gap between statistical leverage scores/effective resistances and the analysis of random graphs by Janson [Combin. Probab. Comput., 3 (1994), pp. 97--126]. This leads to a routine that, in quadratic time, sparsifies a graph down to about $n^{1.5}$ edges in a way that preserves both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $(1 \pm \delta)$ approximation to the determinant of any SDDM matrix with constant probability in about $n^2 \delta^{-2}$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates, in about $n^2 \delta^{-2}$ time, a spanning tree of a weighted undirected graph from a distribution with a total variation distance of $\delta$ from the $\boldsymbol{\mathit{w}}$-uniform distribution.