Approximate Spanning Tree Counting from Uncorrelated Edge Sets

2025-05-20

AI Summary

This work addresses the problem of efficiently approximating the number of spanning trees in sparse graphs. We present the first randomized algorithm achieving a $(1 pm epsilon)$-multiplicative approximation with high probability in $widetilde{O}(m^{1.5}epsilon^{-1})$ time. Unlike prior approaches relying on Schur complements and determinant sparsification, our method is grounded in the Schild–Rao–Srivastava theorem on localization of electrical flows. It iteratively identifies and removes edge subsets contributing negligibly to total effective conductance, thereby adaptively simplifying the graph structure. Integrating electrical flow analysis, local spectral approximation, and randomized sampling, our algorithm significantly improves the accuracy–efficiency trade-off over the previous state-of-the-art bound of $widetilde{O}(m + n^{1.875}epsilon^{-7/4})$. This establishes a new paradigm for spanning tree counting in large-scale sparse graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

We show an $widetilde{O}(m^{1.5} epsilon^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+epsilon)$ factor with high probability, improving on the previous best runtime of $widetilde{O}(m + n^{1.875}epsilon^{-7/4})$ 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].

Problem

• Estimating spanning tree count efficiently
• Improving runtime for sparse graphs
• Using uncorrelated edge sets method

Innovation

• Uses electrical flow localization theorem
• Removes uncorrelated edge sets repeatedly
• Improves runtime to O(m^1.5 ε^-1)