基于通用贪心算法的图稀疏化
Graph Sparsification by Universal Greedy Algorithms
摘要：
图稀疏化是指用一个稀疏图来近似任意的图，在社交网络的简化、最小二乘问题、对称正定线性方程组的数值求解等方面有着广泛的应用。本文受著名的稀疏信号恢复算法正交匹配追踪（OMP）的启发，提出了一种确定性的，贪婪边选择算法称为通用贪婪方法（UGA）的图稀疏化。对于一般的频谱稀疏化问题，例如，正子集选择问题，提出了一种非负的UGA算法，该算法需要$\mathbb{R}^n$时间来找到一个具有正系数的$\frac{1+ε/β}{1-ε/β}$ -谱稀疏子，该稀疏子具有稀疏性$\le\lceil\frac{n}{ε^2}\rceil$，其中$β$是向量的最小长度与最大长度之比.证明了非负UGA算法的收敛性。对于图的稀疏化问题，本文提出了另一种UGA算法，该算法在适当的假设条件下，可以在$O(m+n^2/ε^2)$时间内从一个具有$m$边和$n$顶点的图输出一个具有$\lceil\frac{n}{ε^2}\rceil$边的$\frac{1+O(ε)}{1-O(ε)}$ -谱稀疏因子.这是一个线性时间算法，根据图稀疏化社区正在寻找的边的数量。据作者所知，文献中的最佳结果是存在一个几乎是线性的确定性算法，即对于某些$o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$的$O(m^{1+o(1)})$。最后，广泛的实验结果，包括应用图聚类和最小二乘回归，表明所提出的方法的有效性。
英文摘要：
Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, numerical solution of symmetric positive definite linear systems and etc. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm called universal greedy approach (UGA) for graph sparsification. For a general spectral sparsification problem, e.g., positive subset selection problem from a set of $m$ vectors from $\mathbb{R}^n$, we propose a nonnegative UGA algorithm which needs $O(mn^2+ n^3/ε^2)$ time to find a $\frac{1+ε/β}{1-ε/β}$-spectral sparsifier with positive coefficients with sparsity $\le\lceil\frac{n}{ε^2}\rceil$, where $β$ is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm will be established. For the graph sparsification problem, another UGA algorithm will be proposed which can output a $\frac{1+O(ε)}{1-O(ε)}$-spectral sparsifier with $\lceil\frac{n}{ε^2}\rceil$ edges in $O(m+n^2/ε^2)$ time from a graph with $m$ edges and $n$ vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. $O(m^{1+o(1)})$ for some $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
