Graph Sparsification by Universal Greedy Algorithms

Graph sparsification is to approximate an arbitrary graph by a sparse and useful in many applications, such as simplification of social networks, least squares problems, numerical solution symmetric positive definite linear systems etc. In this paper, inspired the well-known signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce deterministic, greedy edge selection universal approach (UGA) for sparsification. For general spectral problem, e.g., subset problem from set $m$ vectors $\mathbb{R}^n$, propose nonnegative UGA which needs $O(mn^2+ n^3/\epsilon^2)$ time find $\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with coefficients sparsity $\le\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$ ratio between smallest length largest vectors. The convergence will be established. another proposed can output $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral $\lceil\frac{n}{\epsilon^2}\rceil$ edges $O(m+n^2/\epsilon^2)$ $n$ vertices under some mild assumptions. This terms number that community looking for. best result literature knowledge authors existence deterministic almost linear, i.e. $O(m^{1+o(1)})$ $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications clustering regression, show effectiveness approaches.