一种基于SDP的线性尺度谱稀疏化算法
An SDP-Based Algorithm for Linear-Sized Spectral Sparsification
摘要：
对于任何具有$n$顶点和$m$边的无向加权图$G=(V,E,w)$，如果\[ (1-\varepsilon)x^{\intercal}L_Gx\leq x^{\intercal} L_{H} x\leq (1+\varepsilon) x^{\intercal} L_Gx \]对任何$x\in\mathbb{R}^n$成立，则我们称$G$的稀疏子图$H$为$(1+\varepsilon)$ -谱稀疏子，其中$L_G$和$L_{H}$分别是$G$和$H$的拉普拉斯矩阵。注意到任何算法都需要$Ω(m)$时间来构造谱稀疏器，并且$G$的谱稀疏器需要$Ω(n)$边缘，一个自然的问题是调查，对于任何常数$\varepsilon$，是否可以在$\tilde{O}(m)$时间内构造具有$O(n)$边缘的$G$的$(1+\varepsilon)$ -谱稀疏器，其中$\tilde{O}$符号抑制多对数因子。所有以前的频谱稀疏化的建设需要超线性数量的边缘或$m^{1+Ω(1)}$时间。在这项工作中，我们肯定地回答这个问题，提出了一个算法，对于任何无向图$G$和$\varepsilon>0$，输出一个$(1+\varepsilon)$ -频谱稀疏的$G$与$O(n/\varepsilon^2)$边在$\tilde{O}(m/\varepsilon^{O(1)})$时间。我们的算法是基于三个新的技术：（1）一个新的潜在的功能，这是更容易计算，但有类似的保证，在以前的参考文献中使用的潜在的功能;（2）一个有效的减少，从一个双边的谱稀疏到一个单边的谱稀疏;（3）构造一个单边的谱稀疏的半定规划。
英文摘要：
For any undirected and weighted graph $G=(V,E,w)$ with $n$ vertices and $m$ edges, we call a sparse subgraph $H$ of $G$, with proper reweighting of the edges, a $(1+\varepsilon)$-spectral sparsifier if \[ (1-\varepsilon)x^{\intercal}L_Gx\leq x^{\intercal} L_{H} x\leq (1+\varepsilon) x^{\intercal} L_Gx \] holds for any $x\in\mathbb{R}^n$, where $L_G$ and $L_{H}$ are the respective Laplacian matrices of $G$ and $H$. Noticing that $Ω(m)$ time is needed for any algorithm to construct a spectral sparsifier and a spectral sparsifier of $G$ requires $Ω(n)$ edges, a natural question is to investigate, for any constant $\varepsilon$, if a $(1+\varepsilon)$-spectral sparsifier of $G$ with $O(n)$ edges can be constructed in $\tilde{O}(m)$ time, where the $\tilde{O}$ notation suppresses polylogarithmic factors. All previous constructions on spectral sparsification require either super-linear number of edges or $m^{1+Ω(1)}$ time. In this work we answer this question affirmatively by presenting an algorithm that, for any undirected graph $G$ and $\varepsilon>0$, outputs a $(1+\varepsilon)$-spectral sparsifier of $G$ with $O(n/\varepsilon^2)$ edges in $\tilde{O}(m/\varepsilon^{O(1)})$ time. Our algorithm is based on three novel techniques: (1) a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references; (2) an efficient reduction from a two-sided spectral sparsifier to a one-sided spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a semi-definite program.
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