Approximate Spanning Tree Counting from Uncorrelated Edge Sets

TL;DR

This paper introduces a faster algorithm for approximately counting spanning trees in graphs, leveraging electrical flow localization to improve runtime especially in sparse graphs.

Contribution

It presents a novel algorithm that uses uncorrelated edge removal and electrical flow techniques, outperforming previous methods based on Schur complements.

Findings

Achieves $ ilde{O}(m^{1.5} ext{epsilon}^{-1})$ runtime for approximate spanning tree counting.

Improves upon previous algorithms with higher complexity in sparse graphs.

Utilizes electrical flow localization theorem for efficient edge set removal.

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].