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Approximating the Expansion Profile and Almost Optimal Local Graph Clustering
Spectral partitioning is a simple, nearly-linear time, algorithm to find
sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for
the quality of the approximation found by the algorithm. Local graph
partitioning algorithms [ST08,ACL06,AP09] run in time that is nearly linear in
the size of the output set, and their approximation guarantee is worse than the
guarantee provided by the Cheeger inequalities by a polylogarithmic
factor. It has been a long standing open problem to design
a local graph clustering algorithm with an approximation guarantee close to the
guarantee of the Cheeger inequalities and with a running time nearly linear in
the size of the output.
In this paper we solve this problem; we design an algorithm with the same
guarantee (up to a constant factor) as the Cheeger inequality, that runs in
time slightly super linear in the size of the output. This is the first
sublinear (in the size of the input) time algorithm with almost the same
guarantee as the Cheeger's inequality. As a byproduct of our results, we prove
a bicriteria approximation algorithm for the expansion profile of any graph.
Let . There is a polynomial
time algorithm that, for any , finds a set of measure
, and expansion . Our proof techniques also provide a simpler
proof of the structural result of Arora, Barak, Steurer [ABS10], that can be
applied to irregular graphs.
Our main technical tool is that for any set of vertices of a graph, a
lazy -step random walk started from a randomly chosen vertex of , will
remain entirely inside with probability at least . This
itself provides a new lower bound to the uniform mixing time of any finite
states reversible markov chain
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