Sparsification for fast algorithms

In this report we survey some of the recent work on Sparsification of graphs and matrices, and their applications in constructing fast algorithms for linear regression and combinatorial problems such as MaxFlow. We describe state of the art techniques in graph sparsification using matrix sparsification and present without proof some of the previous work which used spectral graph theory. After that we move on to application of those techniques in the specific problem of Multicommodity flow.