Graph Sparsification by Universal Greedy Algorithms
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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 can output a (1+ϵ)^2/(1-ϵ)^2-spectral sparsifier with ⌈n/ϵ^2⌉ edges in O(m+n^2/ϵ^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((loglog(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/ϵ^2) time and the convergence is established.
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