The paper gives an O~(m^1.5 epsilon^-1) time algorithm for approximate spanning tree counting using recursive deletion of uncorrelated edge sets found via electrical-flow localization.
Liu, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang
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Approximate Spanning Tree Counting from Uncorrelated Edge Sets
The paper gives an O~(m^1.5 epsilon^-1) time algorithm for approximate spanning tree counting using recursive deletion of uncorrelated edge sets found via electrical-flow localization.