An $m+n^{3/2}$ algorithm for counting spanning trees by $\ell_1$-regularized resistance

Rong-Hua Li; Yichun Yang

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arXiv · 2609.03574

An $m+n^{3/2}$ algorithm for counting spanning trees by $\ell_1$-regularized resistance

Rong-Hua Li·Yichun Yang

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Abstract

We study the basic problem of approximating the number of spanning trees of a graph. For a graph with $n$ vertices, $m$ edges, We propose an algorithm that approximates the number of spanning trees in $\widetilde O(m+n^{3/2}\eps^{-1})$ time. Our algorithm improves upon the previously best known $\widetilde O(m+n^{15/8}\eps^{-7/4})$ time algorithm by Chu, Gao, Peng, Sachdeva, Sawlani, and Wang [FOCS 2018] and the $\widetilde O(m^{3/2}\eps^{-1})$ time algorithm by Liu, Peng, and Yang [FOCS 2026]. Notably, our algorithm is based on the novel concept of $\ell_1$-regularized resistance. We propose simple and efficient algorithm for computing $\ell_1$-regularized resistance and we show that they can be used to approximate the number of spanning trees by combining with the determinant sparsifier framework of Durfee, Peebles, Peng, and Rao [FOCS 2017]. Our algorithm matches the best known size of the determinant sparsifiers.

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BibTeX RIS

Rong-Hua Li, Yichun Yang. 2026-09-13. An $m+n^{3/2}$ algorithm for counting spanning trees by $\ell_1$-regularized resistance. https://arxiv.org/abs/2609.03574

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