Markov chain methods for small-set expansion
Abstract
Consider a finite irreducible Markov chain with invariant distribution . We use the inner product induced by and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph's Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an "analytic" version of the Small-Set Expansion Problem, which, in turn, immediately gives an approximation algorithm for Small-Set Expansion. We also give a simpler proof of a lower bound on the probability that a random walk stays within a set; this result was used in some recent works on finding small sparse cuts.
Cite
@article{arxiv.1204.4688,
title = {Markov chain methods for small-set expansion},
author = {Ryan O'Donnell and David Witmer},
journal= {arXiv preprint arXiv:1204.4688},
year = {2013}
}