稠密图的近线性时间确定性谱稀疏化
Deterministic Spectral Sparsification in Almost-Linear Time for Dense Graphs
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中文总结 AI 辅助
该研究针对稠密图，提出一种近线性时间的确定性谱稀疏化算法，通过悲观估计器选择匹配与递归阻塞方案，实现了高效的谱稀疏器构造，时间复杂度达到m^(1+o(1))+Õ_ε(n²)。
中文摘要 AI 辅助
带权图的谱稀疏器是一种重权子图，其拉普拉斯二次型可近似原始图的拉普拉斯二次型。设G为正加权n顶点、m边多重图，0<ε≤1/2，假设m、ε⁻¹≤n^O(1)且最大与最小权重之比为多项式有界，我们确定性构造具有O(nε⁻²log^(24+o(1))n)条边的(1±ε)谱稀疏器，耗时为m^(1+o(1))+O(n²ε^(-9/2)log^(113/2+o(1))n)。该构造有两个核心部分：第一，我们通过将近似正则扩展器的边划分为若干匹配，并将其归一化拉普拉斯矩阵视为各向同性半正定矩阵族来实现稀疏化；我们未从该族采样并应用矩阵切尔诺夫界，而是采用悲观估计器确定性选择匹配，通过两种方式评估所得条件期望得分以生成两种算法：使用稠密矩阵乘法，以及稀疏使用逆平方根和矩阵指数的多项式近似。确定性扩展器分解，结合将顶点替换为固定扩展器图以实现近似正则性，可将这些算法扩展至一般图。第二，递归阻塞方案将稠密算法应用于较小子图，将稀疏算法应用于它们的并集，以平衡二者的成本；将所得算法作为稠密算法重复使用，可得α_(r+1)=3-1/(α_r-1)，初始α₀=ω，经O(log n)层后指数为2+O(1/log n)，从而得到m^(1+o(1))+Õ_ε(n²)的时间复杂度。
英文摘要
A spectral sparsifier of a weighted graph is a reweighted subgraph whose Laplacian quadratic form approximates that of the original graph. Let $G$ be a positively weighted $n$-vertex, $m$-edge multigraph, let $0<\varepsilon\le1/2$. Assuming $m,\varepsilon^{-1}\le n^{O(1)}$ and the ratio of maximum to minimum weight is polynomially bounded, we deterministically construct a $(1\pm\varepsilon)$-spectral sparsifier with \[ O\!\left(n\varepsilon^{-2}\log^{24+o(1)}n\right) \] edges in \[ m^{1+o(1)}+O\!\left(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n\right) \] time. The construction has two main ingredients. First, we sparsify an approximately regular expander by partitioning its edges into few matchings and viewing their normalized Laplacians as an isotropic family of positive semidefinite matrices. Rather than sample from this family and apply matrix Chernoff, we select matchings deterministically using a pessimistic estimator. We evaluate the resulting conditional-expectation scores in two ways to produce two algorithms: using dense matrix multiplication and sparsely using polynomial approximations to the inverse square root and matrix exponential. Deterministic expander decomposition, along with replacing vertices by fixed expander graphs to achieve approximate regularity, extends these algorithms to general graphs. Second, a recursive blocking scheme applies the dense algorithm to smaller subgraphs and the sparse algorithm to their union, balancing their costs. Reusing the resulting algorithm as the dense algorithm gives $α_{r+1}=3-1/(α_r-1)$, starting from $α_0=ω$. After $O(\log n)$ levels, the exponent is $2+O(1/\log n)$, yielding $m^{1+o(1)}+\widetilde O_{\varepsilon}(n^2)$ time.
