Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (2017)
Berkeley, California, USA
Oct. 15, 2017 to Oct. 17, 2017
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2017.90
We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores/effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about n1.5 edges in ways that preserve 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 Cholesky 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±δ) approximation to the determinant of any SDDM matrix with constant probability in about n2δ-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 n2δ-2 time a spanning tree of a weighted undirected graph from a distribution with total variation distance of δ from the w-uniform distribution.
approximation theory, computational complexity, matrix algebra, probability, sampling methods, trees (mathematics)
D. Durfee, J. Peebles, R. Peng and A. B. Rao, "Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees," 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, California, USA, 2017, pp. 926-937.