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Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile
We prove two generalizations of the Cheeger's inequality. The first
generalization relates the second eigenvalue to the edge expansion and the
vertex expansion of the graph G, , where
denotes the robust vertex expansion of G and denotes the
edge expansion of G. The second generalization relates the second eigenvalue to
the edge expansion and the expansion profile of G, for all ,
, where denotes the
k-way expansion of G. These show that the spectral partitioning algorithm has
better performance guarantees when is large (e.g. planted random
instances) or is large (instances with few disjoint non-expanding
sets). Both bounds are tight up to a constant factor.
Our approach is based on a method to analyze solutions of Laplacian systems,
and this allows us to extend the results to local graph partitioning
algorithms. In particular, we show that our approach can be used to analyze
personal pagerank vectors, and to give a local graph partitioning algorithm for
the small-set expansion problem with performance guarantees similar to the
generalizations of Cheeger's inequality. We also present a spectral approach to
prove similar results for the truncated random walk algorithm. These show that
local graph partitioning algorithms almost match the performance of the
spectral partitioning algorithm, with the additional advantages that they apply
to the small-set expansion problem and their running time could be sublinear.
Our techniques provide common approaches to analyze the spectral partitioning
algorithm and local graph partitioning algorithms