Spectral Sparsification of Undirected Graphs

A cut sparsifier for an undirected graph is a sparse graph, which preserves up to multiplicative (1 ") factor the values of all cuts. It was shown in [BK96] how to construct cut sparsifiers with O(n logn="2) edges in near-linear time. In [ST11] Spielman and Teng considered much stronger notion: spectral sparsification. A spectral sparsifier is a sparse graph, which (almost) preserves the Laplacian of the original graph. This report (which was written as a final project for the MIT course “Advanced Algorithms”) surveys two constructions of spectral sparsifiers. The first construction is by Spielman and Srivastava [SS11]. They show how to build spectral sparsifiers with O(n logn="2) edges in near-linear time. The second construction is by Batson, Spielman and Srivastava [BSS09]. While having fewer edges (O(n="2) instead of O(n logn="2)), these sparsifiers require much more (but still polynomial) time to construct.