SDDM矩阵的保行列式稀疏化及其在生成树计数和抽样中的应用
Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees
摘要：
我们表明，频谱稀疏化例程的变体可以保持图的总生成树计数，这是基尔霍夫的矩阵树定理，相当于一个图的拉普拉斯小调，或等价的，任何SDDM矩阵的行列式。我们的分析利用这种组合连接来桥接统计杠杆分数/有效阻力和随机图分析[Janson，Combinatorics，Probability and Computing '94]。这导致了一个例程，在二次时间内，以保持行列式和生成树分布的方式将图稀疏化到大约$n^{1.5}$边（假设稀疏化的图被视为随机对象）。扩展该算法的工作与舒尔补充和近似Choleksy因子分解导致算法的计数和采样生成树，这是近最佳的稠密图。我们给出了一个算法，计算一个$(1 \pm δ)$近似的行列式的任何SDDM矩阵具有恒定的概率在约$n^2 δ^{-2}$时间。这是图的第一个例程，它在计算任意矩阵的行列式时优于通用例程。我们还给出了一个算法，在大约$n^2 δ^{-2}$的时间生成的一个加权无向图的生成树从分布的总变化距离为$δ$从$w$ -均匀分布。
英文摘要：
We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about $n^{1.5}$ edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $(1 \pm δ)$ approximation to the determinant of any SDDM matrix with constant probability in about $n^2 δ^{-2}$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates in about $n^2 δ^{-2}$ time a spanning tree of a weighted undirected graph from a distribution with total variation distance of $δ$ from the $w$-uniform distribution .
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