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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 $$n^{1.5}$$ 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 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 total variation distance of $$\delta$$ from the $$w$$-uniform distribution .
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This content will become publicly available on June 24, 2025
Sampling Balanced Forests of Grids in Polynomial Time
We prove that a polynomial fraction of the set of $$k$$-component forests in the $$m \times n$$ grid graph have equal numbers of vertices in each component, for any constant $$k$$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $$k$$-partition according to the product, across its $$k$$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces. Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $$(1 \pm \varepsilon)$$ constant with constant probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. This implies polynomial-time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs. Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $$k$$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.
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- Award ID(s):
- 2104795
- PAR ID:
- 10510434
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing
- Format(s):
- Medium: X
- Location:
- Vancouver, BC, Canada
- Sponsoring Org:
- National Science Foundation
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