Approximate Spanning Tree Counting from Uncorrelated Edge Sets

TL;DR

This work addresses efficiently estimating the spanning-tree count $\mathcal{T}(G)$ of an undirected graph within a $(1+\varepsilon)$ factor. It introduces a novel framework based on repeatedly removing uncorrelated edge subsets derived from electrical-flow localization, and uses a determinant expansion together with a Taylor approximation to relate changes in $\mathcal{T}$ to a small, low-variance estimator on the removed edges. The main result is an algorithm with runtime $\tilde{O}(m^{1.5}\varepsilon^{-1})$ that achieves high-probability $O(\varepsilon)$ additive error in the log-spanning-tree-count, improving over previous near-quadratic runtimes in sparse graphs. This approach avoids Schur-complement sparsification and leverages accurate leverage-score estimation, linear-combination sketching, and $\ell_1$-sketches to enable near-linear-time Laplacian-based determinant estimation with practical implications for related Laplacian computations.

Abstract

We show an $\widetilde{O}(m^{1.5} ε^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+ε)$ factor with high probability, improving on the previous best runtime of $\widetilde{O}(m + n^{1.875}ε^{-7/4})$ in sparse graphs. While previous algorithms were based on computing Schur complements and determinantal sparsifiers, our algorithm instead repeatedly removes sets of uncorrelated edges found using the electrical flow localization theorem of Schild-Rao-Srivastava [SODA 2018].

Figures (1)

• Figure 1: Our recursive algorithm for undirected graph spanning tree approximation

Theorems & Definitions (20)

• Theorem 1
• Lemma 2.1
• proof
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 4.1
• proof
• Lemma 4.2
• proof