  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 $O(mn^2+
n^3/\epsilon^2)$ time to find a
$\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with positive
coefficients with sparsity $\le\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$
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 $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral sparsifier with
$\lceil\frac{n}{\epsilon^2}\rceil$ edges in $O(m+n^2/\epsilon^2)$ 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.
