AI 中文总结
该研究针对图生成树计数问题，基于ℓ₁正则化电阻提出新算法，结合行列式稀疏化框架，在特定边数条件下优于现有算法，提升了生成树计数的时间复杂度。
AI 中文摘要
我们研究图的生成树数量近似这一基本问题。针对含n个顶点、m条边的图，我们提出一种算法，可在Õ(m + n^(7/4)ε^(-3/2))时间内近似生成树数量。当m≥n^(7/6)时，该算法优于Chu、Gao、Peng、Sachdeva、Sawlani与Wang（2018年FOCS会议）提出的Õ(m + n^(15/8)ε^(-7/4))时间算法，以及Liu、Peng与Yang（2026年FOCS会议）提出的Õ(m^(1.5)ε^(-1))时间算法。值得注意的是，我们的算法基于ℓ₁正则化电阻这一新概念，提出了简单高效的ℓ₁正则化电阻计算算法，并结合Durfee、Peebles、Peng与Rao（2017年FOCS会议）的行列式稀疏化框架，证明其可用于近似生成树数量。
英文摘要
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.
