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 algorithm (UGA) for graph sparsification. The UGA algorithm^n, we propose a nonnegative UGA algorithm which needs O(mn^2+ n^3/\epsilon^2) time to find a 1+\epsilon/\beta{1-\epsilon/\beta}-spectral sparsifier with positive coefficients with sparsity }\le\lceil{\epsilon^2}}\rceil can output a \frac{(1+\epsilon)^2{(1-\epsilon)^2}} {1-O(\epsilon)}} -spectral sparsifier with \lceiln{\epsilon^2}\rceil edges in O(m+n^2/\epsilon^2) time  with m edges and n vertices for a general random graph satisfying a mild sufficient condition . 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((\log\log(m))^{2/3}{\log^{1/3}(m)}). We shall point out that several random graphs satisfy the sufficient condition and hence, can be sparsified in linear time. For a general spectral sparsification problem, e.g., positive subset selection problem, a nonnegative UGA algorithm is proposed which needs O(mn^2+ n^3/\epsilon^2) time and the convergence is established .
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 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 {(1-\epsilon)^2}} \frac{1+O(\epsilon){1-O(\epsilon)}} -spectral sparsifier with \lceiln{\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((\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 .
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algorithm
approach
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The UGA algorithm
For a general spectral sparsification problem, e.g., positive subset selection problem from a set of m vectors from \mathbb{R
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\frac{n
A
, 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
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\frac{(1+\epsilon)^2
A
\frac{1+O(\epsilon)
A
from a graph
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for a general random graph satisfying a mild sufficient condition
under some mild assumptions
R
We shall point out that several random graphs satisfy the sufficient condition and hence, can be sparsified in linear time. For a general spectral sparsification problem, e.g., positive subset selection problem, a nonnegative UGA algorithm is proposed which needs O(mn^2+ n^3/\epsilon^2) time and the convergence is established
Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches